$$ \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x \leq \mathrm{Ext}-\int_{a}^{\overline{b}} f(x)d^{\#}x. \qquad (11.69) $$

**Theorem 11.33.** If $f$ is integrable on $[a, b]$, then

$$ \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x = \mathrm{Ext}-\int_{a}^{\overline{b}} f(x)d^{\#}x = \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x. \qquad (11.70) $$

**Theorem 11.34.** If $f$ is bounded (or hyperbounded) on $[a, b]$ and

$$ \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x = \mathrm{Ext}-\int_{a}^{\overline{b}} f(x)d^{\#}x = L, \qquad (11.71) $$

then $f(x)$ is integrable on $[a, b]$ and

$$ \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x = L. \qquad (11.72) $$

**Theorem 11.35.** A bounded (hyperbounded) function $f$ is integrable on $[a, b]$ if and only if

$$ \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x = \mathrm{Ext}-\int_{a}^{\overline{b}} f(x)d^{\#}x. \qquad (11.73) $$

**Theorem 11.36.** If $f$ is bounded (hyperbounded) on $[a, b]$, then $f$ is integrable on $[a, b]$ if and only if for each $\varepsilon > 0$, $\varepsilon \approx 0$ there is a partition $P$ of $[a, b]$ for which

$$ S(P) - s(P) < \varepsilon. \qquad (11.74) $$

**Theorem 11.37.** If $f$ is $\#$-continuous on $[a, b]$, then $f$ is integrable on $[a, b]$.

**Theorem 11.38.** If $f$ is monotonic on $[a, b]$, then $f$ is integrable on $[a, b]$.

**Theorem 11.39.** (a) If $f$ and $g$ are integrable on $[a, b]$, then so is $f+g$, and

$$ \mathrm{Ext}-\int_{a}^{b} [f(x) + g(x)]d^{\#}x = \mathrm{Ext}-\int_{a}^{b} f(x)d^{\#}x + \mathrm{Ext}-\int_{a}^{b} g(x)d^{\#}x \qquad (11.75) $$

(b) If $f_i, 1 \leq i \leq n \in \mathbb{N}^\#$ are integrable on $[a, b]$, then so is $\mathrm{Ext}-\sum_{j=1}^{n} f_i(x)$, and

$$ \mathrm{Ext}-\int_{a}^{b} \left[ \mathrm{Ext}-\sum_{j=1}^{n} f_{i}(x) \right] d^{\#}x = \mathrm{Ext}-\sum_{j=1}^{n} \left( \mathrm{Ext}-\int_{a}^{b} f_{i}(x) d^{\#}x \right) \qquad (11.76) $$

**Theorem 11.40.** (a) If $f$ is integrable on $[a, b]$ and $c \in \mathbb{R}_c^\#$ is a constant, then $cf$ is integrable on $[a, b]$ and

$$ \mathrm{Ext}-\int_{a}^{b} c f(x) d^{\#}x = c \left( \mathrm{Ext}-\int_{a}^{b} f(x) d^{\#}x \right). \qquad (11.77) $$

(b) If $f_i, 1 \leq i \leq n \in \mathbb{N}^\#$ are integrable on $[a, b]$ and $c_i \in \mathbb{R}_c^\#$ are constants, then

$$ \mathrm{Ext}-\int_{a}^{b} \left[ \mathrm{Ext}-\sum_{j=1}^{n} c_i f_i(x) \right] d^{\#}x = \mathrm{Ext}-\sum_{j=1}^{n} c_i \left( \mathrm{Ext}-\int_{a}^{b} f_i(x) d^{\#}x \right). \qquad (11.78) $$

**Theorem 11.41.** If $f$ is and $g$ integrable on $[a, b]$ and $f(x) \leq g(x)$ for $x \in [a, b]$, then
$$ Ext- \int_{a}^{b} f(x) d^{\#}x \le Ext- \int_{a}^{b} g(x) d^{\#}x. \qquad (11.79) $$

**Theorem 11.42.** If $f(x)$ is integrable on $[a, b]$, then so is $|f(x)|$, and

$$ \left| Ext- \int_{a}^{b} f(x) d^{\#}x \right| \le Ext- \int_{a}^{b} |f(x)| d^{\#}x. \qquad (11.80) $$

**Theorem 11.43.** If $f(x)$ and $g(x)$ are integrable on $[a, b]$, then so is the product $f(x)g(x)$.

**Theorem 11.44.** (First Mean Value Theorem for Integrals) Suppose that $u(x)$ is $\#$-continuous and $v(x)$ is integrable and nonnegative on $[a, b]$. Then

$$ Ext- \int_{a}^{b} f(x)v(x)d^{\#}x = v(c) \left( Ext- \int_{a}^{b} f(x)d^{\#}x \right) \qquad (11.81) $$

for some $c \in [a, b]$.

**Theorem 11.45.** If $f(x)$ is integrable on $[a, b]$ and $a \le a_1 < b_1 \le b$, then $f(x)$ is integrable on $[a_1, b_1]$.

**Theorem 11.46.** If $f(x)$ is integrable on $[a, b]$ and $[b, c]$ then $f(x)$ is integrable on $[a, b]$ and

$$ Ext- \int_{a}^{c} f(x) d^{\#}x = Ext- \int_{a}^{b} f(x) d^{\#}x + Ext- \int_{b}^{c} f(x) d^{\#}x. \qquad (11.82) $$

**Theorem 11.47.** If $f(x)$ is integrable on $[a, b]$ and $a \le c \le b$, then the function $F(x)$ defined by

$$ F(x) = Ext- \int_{c}^{x} f(t) d^{\#}t \qquad (11.83) $$

satisfies a Lipschitz condition on $[a, b]$, and is therefore $\#$-continuous on $[a, b]$.

**Theorem 11.48.** If $f(x)$ is integrable on $[a, b]$ and $a \le c \le b$, then $F(x) = Ext- \int_{c}^{x} f(t) d^{\#}t$ is $\#$-differentiable at any point $x_0 \in [a, b]$, where $f(x)$ is $\#$-continuous, with

$$ F^{\#}(x_0) = f(x_0). \qquad (11.84) $$

If $f(x)$ is $\#$-continuous from the right at $a$, then $F_{+}^{\#}(a) = f(a)$. If $f(x)$ is $\#$-continuous from the left at $b$, then $F_{-}^{\#}(b) = f(b)$.

**Theorem 11.49.** Suppose that $F(x)$ is $\#$-continuous on the $\#$-closed interval $[a, b]$ and $\#$-differentiable on the $\#$-open interval $(a, b)$, and $f(x)$ is integrable on $[a, b]$. Suppose also that $F^{\#}(x) = f(x)$, $a < x < b$. Then

$$ Ext- \int_{a}^{b} f(x) d^{\#}x = F(b) - F(a). \qquad (11.85) $$
If $f^{\#}(x)$ is integrable on $[a, b]$, then

$$ \operatorname{Ext-} \int_{a}^{b} f^{\#}(x) d^{\#}x = f(b) - f(a). \qquad (11.86) $$

**Definition 11.22.** A function $F(x)$ is an \#-antiderivative of $f(x)$ on $[a, b]$ if $F(x)$ is \#-continuous on $[a, b]$ and \#-differentiable on $[a, b]$, with $F^{\prime\#}(x) = f(x)$, $a < x < b$.

**Theorem 11.50.** If $F(x)$ is an \#-antiderivative of $f(x)$ on $[a, b]$, then so is $F(x) + c$ for any constant $c$. Conversely, if $F_1(x)$ and $F_2(x)$ are \#-antiderivatives of $f$ on $[a, b]$, then $F_1(x) - F_2(x)$ is constant on $[a, b]$.

**Theorem 11.51.** (Fundamental Theorem of Calculus) If $f(x)$ is \#-continuous on $[a, b]$, then $f(x)$ has an \#-antiderivative on $[a, b]$. Moreover, if $F(x)$ is any \#-antiderivative of $f$ on $[a, b]$, then

$$ \operatorname{Ext-} \int_{a}^{b} f(x) d^{\#}x = F(b) - F(a). \qquad (11.87) $$

**Theorem 11.52.** (Integration by Parts) If $u^{\prime\#}(x)$ and $v^{\prime\#}(x)$ are integrable on $[a, b]$, then

$$ \operatorname{Ext-} \int_{a}^{b} u(x) v^{\prime\#}(x) d^{\#}x = u(x) v(x) \Big|_{a}^{b} - \operatorname{Ext-} \int_{a}^{b} u(x)^{\prime\#} v(x) d^{\#}x. \qquad (11.88) $$

**Theorem 11.53.** Suppose that the transformation $x = \varphi(t)$ maps the interval $c \le t \le d$ into the interval $a \le x \le b$, with $\varphi(c) = \alpha$ and $\varphi(d) = \beta$, and let $f(x)$ be \#-continuous on $[a, b]$. Let $\varphi^{\prime\#}(t)$ be integrable on $[c, d]$. Then

$$ \operatorname{Ext-} \int_{a}^{\beta} f(x) d^{\#}x = \operatorname{Ext-} \int_{c}^{d} f(\varphi(t)) \varphi^{\prime\#}(t) d^{\#}t. \qquad (11.89) $$

**Theorem 11.54.** Suppose that $\varphi^{\prime\#}(t)$ is integrable and $\varphi(t)$ is monotonic on $[c, d]$, and the transformation $x = \varphi(t)$ maps $[c, d]$ onto $[a, b]$. Let $f(x)$ be bounded (hyperbounded) on $[a, b]$. Then $g(t) = f(\varphi(t)) \varphi^{\prime\#}(t)$ is integrable on $[c, d]$ if and only if $f(x)$ is integrable over $[a, b]$, and in this case

$$ \operatorname{Ext-} \int_{a}^{b} f(x) d^{\#}x = \operatorname{Ext-} \int_{c}^{d} f(\varphi(t)) \varphi^{\prime\#}(t) d^{\#}t. \qquad (11.90) $$

## 11.14. Improper integrals.

**Definition 11.22.** We say $f(x)$ is locally integrable on an interval $I$ if $f(x)$ is integrable on every finite or hyperfinite \#-closed subinterval of $I$.

**Definition 11.23.** If $f$ is locally integrable on $[a, b]$, we define

$$ \operatorname{Ext-} \int_{a}^{b} f(x) d^{\#}x = \operatorname{\#-lim}_{c \to a, c \neq b} \left( \operatorname{Ext-} \int_{a}^{c} f(x) d^{\#}x \right). \qquad (11.91) $$

**Remark 11.3.** The \#-limit in (91.91) always exists if $[a, b]$ is finite or hyperfinite and $f$ is locally integrable and bounded (hyperbounded) on $[a, b]$. In this case, Definitions 11.70 and 11.91 assign the same value to $\operatorname{Ext-} \int_{a}^{b} f(x) d^{\#}x$ no matter how $f$
is defined. However, the #-limit may also exist in cases where $b = \infty^{\#}$ or $b < \infty^{\#}$ and $f$ is hyper unbounded as $x$ approaches $b$ from the left.

**Definition 11.24.** In the cases mentioned above, Definition 11.91 assigns a value to an integral that does not exist in the sense of Definition 11.70, and $Ext-\int_a^b f(x)d^\#x$ is said to be an *improper integral* that #-converges to the #-limit in (11.91). We also say in this case that $f$ is integrable on $[a, b]$ and that $Ext-\int_a^b f(x)d^\#x$ exists.

If the #-limit in (11.91) does not exist (finite or hyperfinite), we say that the improper integral $Ext-\int_a^b f(x)d^\#x$ #-diverges, and $f$ is nonintegrable on $[a, b)$. In particular, if $\#\lim_{c \to a} b- (Ext-\int_a^c f(x)d^\#x) = \pm\infty^\#$ we say that #-diverges to $\infty^\#$, and we write

$$ Ext-\int_a^b f(x)d^\#x = \infty^\# \qquad (11.92) $$

or

$$ Ext-\int_a^b f(x)d^\#x = -\infty^\#, \qquad (11.93) $$

whichever the case may be. Similar comments apply to the next two definitions.

**Definition 11.25.** If $f(x)$ is locally integrable on $(a, b]$, we define

$$ Ext-\int_a^b f(x)d^\#x = \#\lim_{c \to a} \left( Ext-\int_c^b f(x)d^\#x \right). \qquad (11.94) $$

provided that the #-limit exists (finite or hyperfinite). To include the case where $a = -\infty^\#$, we adopt the convention that $-\infty^\# \mp = -\infty^\#$.

**Definition 11.26.** If $f(x)$ is locally integrable on $(a, b)$, we define

$$ Ext-\int_a^b f(x)d^\#x = Ext-\int_a^\alpha f(x)d^\#x + Ext-\int_a^b f(x)d^\#x, \qquad (11.95) $$

where $a < \alpha < b$, provided that both improper integrals on the right exist i.e., finite or hyperfinite.

**Remark 11.4.** Note that the existence and value of $Ext-\int_a^b f(x)d^\#x$ according to Definition 11.26 do not depend on the particular choice of $\alpha \in (a, b)$.

**Remark 11.5.** When we wish to distinguish between improper integrals and integrals in the sense of Definition 11.70, we will call the latter proper integrals.

**Theorem 11.55.** Suppose that $f_1, f_2, \dots, f_n$ are locally integrable on $[a, b)$ and that $Ext-\int_a^b f_1(x)d^\#x, \dots, Ext-\int_a^b f_n(x)d^\#x$ #-converge. Let $c_1, c_2, \dots, c_n$ be constants. Then $Ext-\int_a^b (Ext-\sum_{i=1}^n c_i f_i(x))d^\#x$ #-converges and

$$ Ext-\int_a^b (Ext-\sum_{i=1}^n c_i f_i(x)) d^\#x = Ext-\sum_{i=1}^n c_i (Ext-\int_a^b f_i(x) d^\#x). \qquad (11.96) $$

## 11.15. Improper integrals of nonnegative functions
$f: D \to \mathbb{R}^\#$. Absolute Integrability.

**Theorem 11.56.** If $f(x)$ is nonnegative and locally integrable on $[a, b)$, then
$Ext-\int_a^b f(x)d^\#x$ converges if the function

$$F(x) = Ext-\int_a^x f(x)d^\#x \qquad (11.97)$$

is bounded (hyperbounded) on $[a, b)$, and $Ext-\int_a^b f(x)d^\#x = \infty^\#$ if it is not.

**Theorem 11.57.** (Comparison Test) If $f$ and $g$ are locally integrable on $[a, b)$ and $0 \le f(x) \le g(x)$, $0 \le x < b$, then (a) $Ext-\int_a^b f(x)d^\#x < \infty^\#$ if $Ext-\int_a^b g(x)d^\#x < \infty^\#$ and (b) $Ext-\int_a^b f(x)d^\#x = \infty^\#$ if $Ext-\int_a^b g(x)d^\#x = \infty^\#$.

**Theorem 11.58.** Suppose that $f$ and $g$ are locally integrable on $[a, b)$, $g(x) > 0$ and $f(x) \ge 0$ on some subinterval $[a_1, b) \subset [a, b)$, and

$$\#\lim_{c \to b^-} \frac{f(x)}{g(x)} = M. \qquad (11.98)$$

(a) If $0 < M < \infty^\#$, then $Ext-\int_a^b f(x)d^\#x$ and $Ext-\int_a^b g(x)d^\#x$ converge or diverge together.

(b) If $M = \infty^\#$ and $Ext-\int_a^b g(x)d^\#x = \infty^\#$, then $Ext-\int_a^b f(x)d^\#x = \infty^\#$.

(c) If $M = 0$ and $Ext-\int_a^b g(x)d^\#x < \infty^\#$, then $Ext-\int_a^b f(x)d^\#x < \infty^\#$.

**Definition 11.27.** We say that $f$ is absolutely integrable on $[a, b)$ if $f$ is locally integrable on $[a, b)$ and $Ext-\int_a^b |f(x)|d^\#x < \infty^\#$. In this case we also say that $Ext-\int_a^b f(x)d^\#x$ #-converges absolutely or is absolutely #-convergent.

**Theorem 11.59.** If $f$ is locally integrable on $[a, b)$ and $Ext-\int_a^b |f(x)|d^\#x < \infty^\#$, then $Ext-\int_a^b f(x)d^\#x$ #-converges: that is, an absolutely #-convergent integral is #-convergent.

**Theorem 11.60.** (Dirichlet's Test) Suppose that $f$ is #-continuous and its #-antiderivative $F(x) = Ext-\int_a^x f(x)d^\#x$ is bounded (hyperbounded) on $[a, b)$. Let $g^\#$ be absolutely integrable on $[a, b)$, and suppose that

$$\#\lim_{c \to b^-} g(x) = 0. \qquad (11.99)$$

Then $Ext-\int_a^x f(x)g(x)d^\#x$ #-converges.

**Theorem 11.61.** Suppose that $u(x)$ is #-continuous on $[a, b)$ and $Ext-\int_a^x u(x)d^\#x$ #-diverges. Let $v(x)$ be positive and #-differentiable on $[a, b)$, and suppose that $\#\lim_{c \to b^-} v(x) = \infty^\#$ and $v^\#/v^2$ is absolutely integrable on $[a, b)$. Then $Ext-\int_a^x u(x)v(x)d^\#x$ #-diverges.

**Theorem 11.62.** Suppose that $g(x)$ is monotonic on $[a, b)$ and $Ext-\int_a^b f(x)d^\#x = \infty^\#$. Let $f(x)$ be locally integrable on $[a, b)$ and

$$Ext-\int_{x_j}^{x_{j+1}} |f(x)|d^\#x \ge \rho, j \ge 0 \qquad (11.100)$$

for some positive $\rho$ where $\{x_j\}_{j \in \mathbb{N}^\#}$ is an increasing hyper infinite sequence of points in $[a, b)$ such that $\#\lim_{j \to \infty} x_j = b$ and $x_{j+1}x_j \le M, j \ge 0$, for some $M$. Then

$$Ext-\int_a^b f(x)g(x)|d^\#x| = \infty^\# \qquad (11.101)$$
## 11.16. Change of Variable in an Improper Integral

**Theorem 11.63.** Suppose that $\varphi(t)$ is monotonic and $\varphi^{\prime\#}(t)$ is locally integrable on either of the half-open intervals $I = [c, d)$ or $(c, d]$, and let $x = \varphi(t)$ map $I$ onto either of the half-open intervals $J = [a, b)$ or $J = (a, b]$. Let $f$ be locally integrable on $J$. Then the improper integrals

$$ \operatorname{Ext}-\int_{a}^{b} f(x) d^{\#}x \quad \text{and} \quad \operatorname{Ext}-\int_{a}^{b} f(\varphi)|\varphi^{\prime\#}(t)| d^{\#}t \qquad (11.102) $$

\#-diverge or \#-converge together, in the latter case to the same value. The same conclusion holds if $\varphi(t)$ and $\varphi^{\prime\#}(t)$ have the stated properties only on the \#-open interval $(a, b)$, the transformation $x = \varphi(t)$ maps $(c, d)$ onto $(a, b)$, and $f$ is locally integrable on $(a, b)$.

## 11.17. Generalized integrability criterion due to Lebesgue.

The main result of this section is an integrability criterion due to Lebesgue that does not require computation, but has to do with how badly \#-discontinuous a function may be and still be integrable.

**Definition 11.28.** If $f(x)$ is bounded (hyperbounded) on $[a, b]$, the oscillation $W_f[a, b]$ of $f(x)$ on $[a, b]$ is defined by

$$ W_f[a, b] = \sup_{a \le x, x' \le b} |f(x) - f(x')| \qquad (11.103) $$

which can also be written as

$$ W_f[a, b] = \sup_{a \le x \le b} f(x) - \inf_{a \le x \le b} f(x). \qquad (11.104) $$

**Definition 11.29.** If $a < x < b$, the oscillation $w_f(x)$ of $f(x)$ at $x$ is defined by

$$ w_f(x) = \#\lim_{h \to \# 0^+} W_f(x - h, x + h) \qquad (11.105) $$

The corresponding definitions for $x = a$ and $x = b$ are

$$ w_f(a) = \#\lim_{h \to \# 0^+} W_f(a, a + h) \quad \text{and} \quad w_f(b) = \#\lim_{h \to \# 0^+} W_f(b - h, b). \qquad (11.106) $$

Note that for a fixed $x \in (a, b)$, $W_f(x - h, x + h)$ is a nonnegative and nondecreasing function of $h$ for $0 < h < \min\{x - a, b - x\}$, therefore, $w_f(x)$ exists and is nonnegative.

**Theorem 11.64.** Let $f$ be defined on $[a, b]$. Then $f$ is \#-continuous at $x_0 \in [a, b]$ if and only if $w_f(x) = 0$; \#-continuity at $a$ or $b$ means \#-continuity from the right or left, respectively.

**Definition 11.30.** A subset $S$ of the $\mathbb{R}^\#_c$ is of Lebesgue measure zero if for every $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hyperfinite or hyper infinite sequence of open intervals $I_1, I_2, \dots$ such that

$$ S \subset \bigcup_{j} I_j \qquad (11.107) $$

and

$$ \operatorname{Ext}-\sum_{j=1}^{n} L(I_j) < \varepsilon, n \ge 1. \qquad (11.108) $$

Note that any subset of a set of Lebesgue measure zero is also of Lebesgue measure zero.

**Example 1.** Any hyperfinite set $S = \{x_i\}_{i \in \mathbb{N}}$, $\mathbb{N}^\#\mathbb{N}$ is of Lebesgue measure zero,
since we can choose #-open intervals $I_1, I_2, \dots, I_n$ such that $x_j \in I_j$ and $L(I_j) < \varepsilon/\mathbf{n}$, $1 \le j \le \mathbf{n}$.

**Definition 11.31.** An infinite set $S \subset \mathbb{R}_c^\#$ is hyper denumerable if its members can be listed in a hyper infinite sequence (that is, in a one-to-one correspondence with the positive hyper integers); thus, $S = \{x_i\}_{i \in \mathbb{N}^\#}$. An infinite set that does not have this property is hyper non hyper denumerable.

**Example 2.** Any denumerable set $S = \{x_i\}_{i \in \mathbb{N}^\#}$ is of Lebesgue measure zero, since if $\varepsilon > 0, \varepsilon \approx 0$, it is possible to choose open intervals

$I_1, I_2, \dots$, so that $x_j \in I_j$ and $L(I_j) < 2^{-j}\varepsilon, j \ge 1$. Then (11.108) holds since

$$ \mathrm{Ext}-\sum_{j=1}^{n} 2^{-j} = 1 - 2^{-\mathbf{n}} < 1. $$

**Theorem 11.64.** If $w_f(x) < \varepsilon, \varepsilon \approx 0$, for $a \le x \le b$, then there is a $\delta > 0, \delta \approx 0$ such that $W_f[a, b] < \varepsilon$, provided that $a_1, b_1 \subset [a, b]$ and $b_1 - a_1 < \delta$.

**Theorem 11.65.** Let $f$ be bounded (hyperbounded) on $[a, b]$ and define

$E_\rho = \{x \in [a, b] | w_f(x) > \rho\}$. Then $E_\rho$ is #-closed; and $f$ is integrable on $[a, b]$ if and only if for every pair of positive numbers $\rho$ and $\delta$, $E_\rho$ can be covered by hyper finitely many open intervals $I_1, I_2, \dots, I_\mathbf{p}, \mathbf{p} \in \mathbb{N}^\# \setminus \mathbb{N}$ such that

$$ \mathrm{Ext}-\sum_{j=1}^{\mathbf{p}} L(I_j) < \delta. \qquad (11.109) $$

**Theorem 11.66.** A bounded (hyperbounded) function $f$ is integrable on a finite or hyperfinite interval $[a, b]$ if and only if the set $S$ of #-discontinuities of $f$ in $[a, b]$ is of Lebesgue measure zero.

# 12. Hyper infinite external sequences and series

## 12.1. Hyper infinite external sequences

An hyper infinite sequence (or hypersequence) of $\mathbb{R}_c^\#$-real numbers is a $\mathbb{R}_c^\#$-valued function defined on a set of hyperintegers $\{n | n \in \mathbb{N}^\# \wedge n \ge k \in \mathbb{N}\}$. We call the values of the function the terms of the hypersequence. We denote a hypersequence by listing its terms in order; thus, $\{s_n\}_k^{\infty} = \{s_k, s_{k+1}, \dots\}$. We often write $\{s_n\}_{n \in \mathbb{N}^\#}$ or simple $\{s_n\}$ for a shot.

**Definition 12.1.** A hyper infinite sequence $\{s_n\}_k^{\infty}$ converges to a limit $s \in \mathbb{R}_c^\#$ if for every $\varepsilon \approx 0, \varepsilon > 0$ there is an hyperinteger $N \in \mathbb{N}^\# \setminus \mathbb{N}$ such that

$$ |s_n - s| < \varepsilon \text{ if } n \ge N \qquad (12.1) $$

In this case we say that $\{s_n\}$ is #-convergent and write

$$ \#-\lim_{n \to \infty} \infty^\# s_n = s. \qquad (12.2) $$

A hyper infinite sequence that does not #-converge diverges, or is #-divergent.

**Theorem 12.1.** The #-limit of a #-convergent hypersequence is unique:

**Proof.** Suppose that $\#\lim_{n \to \infty} \infty^\# s_n = s_1$ and $\#\lim_{n \to \infty} \infty^\# s_n = s_2$. We must show that $s = s'$. Let $\varepsilon \approx 0, \varepsilon > 0$. From Definition 10.1, there are hyperintegers $N_1$ and $N_2$ such that $|s_n - s_1| < \varepsilon$ if $n \ge N_1$, and $|s_n - s_2| < \varepsilon$ if $n \ge N_2$. These inequalities both hold if $n \ge N = \max(N_1, N_2)$, which implies that: $|s_1 - s_2| < 2\varepsilon$. Since this inequality holds for every $\varepsilon \approx 0, \varepsilon > 0$ and $|s_1 - s_2|$ is independent of $\varepsilon$, we conclude that $|s_1 - s_2| = 0$; that
is, $s_1 = s_2$.

**Definition 12.2.** A hypersequence {$s_n$} is bounded above if there is a hyperreal number

$b \in \mathbb{R}_c^\#$ such that $s_n \le b$ for all $n \in \mathbb{N}^\#$; bounded below if there is a real number $a \in \mathbb{R}_c^\#$ such that $s_n \ge a$ for all $n \in \mathbb{N}^\#$; or bounded if there is a real number $r \in \mathbb{R}_c^\#$ such that $|s_n| \le r$ for all $n \in \mathbb{N}^\#$.

**Theorem 12.2.** Any #-convergent hypersequence {$s_n$} is bounded or hyperbounded.

**Proof.** By taking $\varepsilon = 1$ in Eq. (12.1), we see that if #-lim$_{n \to \#} \infty^\# s_n = s$, then there is an hyperinteger $N \in \mathbb{N}^\#\mathbb{N}$ such that $|s_n - s| < 1$ if $n \ge N$. Therefore,

$s_n = |(s_n - s) + s| \le |s_n - s| + |s| < 1 + |s|$ if $n \ge N$; and
$s_n \le \max\{(\max_{1 \le i \le N-1}\{|s_0|, |s_1|, \dots, |s_{N-1}|\}), 1 + |s|\}$ for all $n \in \mathbb{N}^\#$, so {$s_n$} is bounded.

**Definition 12.3.** (Sequences Diverging to $\pm\infty$#). We say that

$$ \#\text{-lim}_{n \to \#} \infty^\# s_n = +\infty^\# $$

if for any hyperreal number $a, s_n > a$ for any $n \ge N \in \mathbb{N}^\#\mathbb{N}$. Similarly,

$$ \#\text{-lim}_{n \to \#} \infty^\# s_n = -\infty^\# $$

if for any hyperreal number $a, s_n < a$ for any $n \ge N \in \mathbb{N}^\#\mathbb{N}$. However, we do not regard {$s_n$} as #-convergent unless $\#\text{-lim}_{n \to \#} \infty^\# s_n$

is finite or hyperfinite, as required by Definition 12.1. To emphasize this distinction, we say that {$s_n$} diverges to $\infty^#(-\infty^#)$ if $\#\text{-lim}_{n \to \#} \infty^\# s_n = \infty^#(-\infty^#)$.

**Theorem 12.3.** Assume that a nonempty set $S \subset \mathbb{R}_c^\#$ of real $\mathbb{R}_c^\#$-numbers has a supremum $\sup(S)$, then $\sup S$ is the unique hyperreal number $\beta \in \mathbb{R}_c^\#$ such that

(a) $x \le \beta$ for all $x \in S$

(b) if $\varepsilon > 0, \varepsilon \approx 0$ (no matter how infinite small) there is an $x_0 \in S$ such that $x_0 > \beta - \varepsilon$.

**Proof.** We first show that $\beta = \sup S$ has properties (a) and (b). Since $\beta$ is an upper bound of $S$, it must satisfy (a). Since any hyperreal number $\alpha$ less than $\beta$ can be written as $\alpha = \beta - \varepsilon$ with $\varepsilon = \beta - \alpha > 0$, (b) is just another way of saying that no number less than $\beta$ is an upper bound of $S$. Hence, $\beta = \sup S$ satisfies (a) and (b). Now we show that there cannot be more than one hyperreal number with properties (a) and (b).

Suppose that $\beta_1 < \beta_2$ and $\beta_2$ has property (b); thus, if $\varepsilon > 0$, there is an $x_0 \in S$ such that $x_0 > \beta_2 - \varepsilon$. Then, by taking $\varepsilon = \beta_2 - \beta_1$, we see that there is an $x_0 \in S$ such that $x_0 > \beta_2 - (\beta_2 - \beta_1) = \beta_1$, so $\beta_1$ cannot have property (a). Therefore, there cannot be more than one hyperreal number that satisfies both (a) and (b).

**Definition 12.4.** A hypersequence {$s_n$}$_{n \in \mathbb{N}^\#$} is nondecreasing if $s_n \ge s_{n-1}$ for all $n \in \mathbb{N}^\#$, or nonincreasing if $s_n \le s_{n-1}$ for all $n \in \mathbb{N}^\#$. A monotonic hyper infinite sequence is a hyper infinite sequence that is either nonincreasing or nondecreasing. If $s_n > s_{n-1}$ for all $n \in \mathbb{N}^\#$, then {$s_n$}$_{n \in \mathbb{N}^\#$} is increasing, while if $s_n < s_{n-1}$ for all $n \in \mathbb{N}^\#$, {$s_n$}$_{n \in \mathbb{N}^\#$} is decreasing.

**Theorem 12.4.** (a) If {$s_n$}$_{n \in \mathbb{N}^\#$} is nondecreasing and there exists $\sup\{s_n|n \in \mathbb{N}^\#\}$ then
$$ \#\text{-lim}_{n \to \#} \infty^\# s_n = \sup\{s_n|n \in \mathbb{N}^\#\}. $$

(b) If {$s_n$}$_{n \in \mathbb{N}^\#$} is nonincreasing and there exists $\inf\{s_n|n \in \mathbb{N}^\#\}$ then
$$ \#\text{-lim}_{n \to \#} \infty^\# s_n = \inf\{s_n|n \in \mathbb{N}^\#\}. $$

**Proof.** (a) Let $\beta = \sup\{s_n|n \in \mathbb{N}^\#\}$. . If $\beta < +\infty^\#$, Theorem 12.3 implies that if $\varepsilon > 0$ then $\beta - \varepsilon < s_N \le \beta$ for some hyperinteger $N \in \mathbb{N}^\#\mathbb{N}$. Since $s_N \le s_n \le \beta$ if $n \ge N$, it follows
that $\beta - \varepsilon < s_n \le \beta$ if $n \ge N$. This implies that $|s_n - \beta| < \varepsilon$ if $n \ge N$, so $\#\text{-lim}_{n \to \#} \infty^\# s_n = \beta$, by definition of the $\#$-limit. If $\beta = +\infty^\#$

and $b$ is any hyperreal number, then $s_N > b$ for some hyperinteger $N$. Then $s_n > b$ for $n \ge N$, so $\#\text{-lim}_{n \to \#} \infty^\# s_n = +\infty^\#$.

**Theorem 12.5. (Generalized Nested Intervals Theorem)**

Let $\{I_n\}_{n \in \mathbb{N}^\#} = \{[a_n, b_n]\}_{n \in \mathbb{N}^\#}$, $[a_n, b_n] \subset \mathbb{R}_c^\#$ be a hyper infinite sequence of $\#$-closed intervals satisfying each of the following conditions:

(i) $I_1 \supseteq I_2 \supseteq I_3 \supseteq \dots \supseteq I_n \supseteq \dots$,

(ii) $b_n - a_n \to \# 0$ as $n \to \infty^\#$.

Then $\cap_{n=1}^{\infty^\#} I_n$ consists of exactly one hyperreal number $x \in \mathbb{R}_c^\#$. Moreover both hyper infinite sequences $\{a_n\}$ and $\{b_n\}$ $\#$-converge to $x$.

**Proof.** See proof to Theorem 8.11.

**Theorem 12.6. (Generalized Bolzano-Weierstrass Theorem)** Every bounded (hyperbounded) hyper infinite sequence $\{s_n\}_{n \in \mathbb{N}^\#}$ has a $\#$-convergent sub hyper infinite sequence.

**Proof.** Let $\{s_n\}_{n \in \mathbb{N}^\#}$ be a bounded hyper infinite sequence. Then, there exists an interval $[a_1, b_1]$ such that: (i) $a_1, b_1 \in \mathbb{Q}^\#$ and (ii) $a_1 \le s_n \le b_1$ for all $n \in \mathbb{N}^\#$.

Either $[a_1, \frac{a_1+b_1}{2}]$ or $[\frac{a_1+b_1}{2}, b_1]$ contains hyperinfinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$.

That is, there exists hyper infinitely many $n \in \mathbb{N}^\#$ such that $a_n$ is in $[a_1, \frac{a_1+b_1}{2}]$, or there exists hyper infinitely many $n \in \mathbb{N}^\#$ such that $a_n$ is in $[\frac{a_1+b_1}{2}, b_1]$.

If $[a_1, \frac{a_1+b_1}{2}]$ contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$, let $[a_2, b_2] = [a_1, \frac{a_1+b_1}{2}]$. Otherwise, let $[a_2, b_2] = [\frac{a_1+b_1}{2}, b_1]$.

Either $[a_2, \frac{a_2+b_2}{2}]$ or $[\frac{a_2+b_2}{2}, b_2]$ contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$.

If $[a_2, \frac{a_2+b_2}{2}]$ contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$, let $[a_3, b_3] = [a_2, \frac{a_2+b_2}{2}]$. Otherwise, let $[a_3, b_3] = [\frac{a_2+b_2}{2}, b_3]$.

By hyper infinite induction, we can continue this construction and obtain a hyper infinite sequence of intervals $\{[a_n, b_n]\}_{n \in \mathbb{N}^\#}$ such that:

(i) for each $n \in \mathbb{N}^\#$, interval $[a_n, b_n]$ contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$,

(ii) for each $n \in \mathbb{N}^\#$, $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$ and

(iii) for each $n \in \mathbb{N}^\#$, $b_{n+1} - a_{n+1} = \frac{1}{2}(b_n - a_n)$.

The nested intervals theorem implies that the intersection $\bigcap_{n \in \mathbb{N}^\#} [a_n, b_n]$ of all of the intervals $[a_n, b_n]$ is a single point $s$. We will now construct a sub hyper infinite sequence of

$\{s_n\}_{n \in \mathbb{N}^\#}$ which will $\#$-converge to $s$.

Since $[a_1, b_1]$ contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$, there exists $k_1 \in \mathbb{N}^\#$ such that $s_{k_1}$ is in $[a_1, b_1]$. Since $[a_2, b_2]$ contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$, there exists $k_2 \in \mathbb{N}^\#$, $k_2 > k_1$ such that $s_{k_2}$ is in $[a_2, b_2]$. Since $[a_3, b_3]$

contains hyper infinitely many terms of $\{s_n\}_{n \in \mathbb{N}^\#}$, there exists $k_3 \in \mathbb{N}^\#$, $k_3 > k_2$ such that $s_{k_3}$ is in $[a_3, b_3]$. Continuing this process by hyper infinite induction, we obtain a hyper infinite sequence $\{s_{k_n}\}_{n \in \mathbb{N}^\#}$ such that $s_{k_n} \in [a_n, b_n]$ for each $n \in \mathbb{N}^\#$. The hypersequence $\{s_{k_n}\}_{n \in \mathbb{N}^\#}$ is a sub hyper infinite sequence of $\{s_n\}_{n \in \mathbb{N}^\#}$ since $k_{n+1} > k_n$ for each $n \in \mathbb{N}^\#$. Since $\#\text{-lim}_{n \to \infty^\#} a_n = s$ and $\#\text{-lim}_{n \to \infty^\#} b_n = s$ and $a_n \le s_n \le b_n$ for each $n \in \mathbb{N}^\#$, the squeeze theorem implies that that $\#\text{-lim}_{n \to \infty^\#} s_n = s$.
## 12.2. Hyper infinite external series of constant.

**Definition 12.5.** If {$a_n$}$_k^{\infty}$ is an hyper infinite external sequence of Cauchy hyperreal numbers, the symbol

$$ \mathrm{Ext}-\sum_{n=k}^{\infty} a_n \qquad (12.3) $$

is an hyper infinite series, and $a_n$ is the n-th term of the hyper infinite series.

We say that $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$ #-converges to the sum $A \in \mathbb{R}_c^\#$, and write

$$ \mathrm{Ext}-\sum_{n=k}^{\infty} a_n = A \qquad (12.4) $$

if the hyper infinite sequence {$A_n$}$_k^{\infty}$ defined by

$$ A_n = \mathrm{Ext}-\sum_{i=k}^{i=n} a_n \qquad (12.5) $$

$n \in \mathbb{N}^\#$, #-converges to A. The hyper infinite sum $A_n$ is the n-th partial sum of $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$

If {$A_n$}$_k^{\infty}$ diverges, we say that $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$ diverges; in particular, if $\lim_{n \to \infty} A_n = \infty^\#$

or $-\infty^\#$, we say that $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$ #-diverges to $\infty^\#$ or $-\infty^\#$, and write

$$ \mathrm{Ext}-\sum_{n=k}^{\infty} a_n = \infty^\# \text{ or } \mathrm{Ext}-\sum_{n=k}^{\infty} a_n = -\infty^\#. \qquad (12.6) $$

A divergent hyper infinite series that does not diverge to $\pm\infty^\#$ is said to oscillate, or be oscillatory.

**Example 12.1** Consider the hyper infinite series

$$ \mathrm{Ext}-\sum_{n=0}^{\infty} r^n, -1 < r < 1. \qquad (12.7) $$

Here $a_n = r^n, n \ge 0, n \in \mathbb{N}^\#$ and

$$ A_n = 1 + r + r^2 + \dots + r^n = \frac{1 - r^{n+1}}{1 - r} \qquad (12.8) $$

which #-converges to $1 = 1/(1-r)$ as $n \to \infty^\#$; thus, we write

$$ \mathrm{Ext}-\sum_{n=0}^{\infty} r^n = 1/(1-r), -1 < r < 1. $$

An hyper infinite series can be viewed as a generalization of a hyperfinite sum

$$ A_N = \mathrm{Ext}-\sum_{n=k}^{N} a_n \qquad \text{Therefore, } \#\mathrm{lim}_{N \to \infty} A_N = A. $$

**Theorem 12.7.** The sum of a #-convergent hyper infinite series is unique:
**Theorem 12.8.** Let $\sum_{n=k}^{\infty} a_n = A$ and $\sum_{n=k}^{\infty} b_n = B$ where A and B are finite or hyperfinite.

Then

$$ \mathrm{Ext}-\sum_{n=k}^{\infty} (a_n \pm b_n) = A \pm B \qquad (12.9) $$

and

$$ \mathrm{Ext}-\sum_{n=k}^{\infty} (c \times a_n) = c \times A \qquad (12.10) $$

if $c \in \mathbb{R}_c^\#$ is a constant.

**Theorem 12.9.** (Cauchy's #-convergence criterion for hyper infinite series) A hyper infinite series $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$ #-converges if and only if for every $\varepsilon > 0, \varepsilon \approx 0$ there is an hyperinteger $N \in \mathbb{N}^\# \setminus \mathbb{N}$ such that

$$ |\mathrm{Ext}-\sum_{n}^{m} a_n| < \varepsilon \qquad (12.11) $$

if $m \ge n \ge N$.

**Corollary 12.1.** If $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$ #-converges; then #-lim$_{N \to \infty}$ $a_n = 0$.

**Corollary 12.2.** If $\mathrm{Ext}-\sum_{n}^{\infty} a_n$ #-converges; then for each $\varepsilon > 0, \varepsilon \approx 0$ there is an hyperinteger $K \in \mathbb{N}^\# \setminus \mathbb{N}$ such that $|\mathrm{Ext}-\sum_{n=k}^{\infty} a_n| < \varepsilon$ if $k \ge K$, that is

$$ \#\lim_{k \to \infty} \# (\mathrm{Ext}-\sum_{n=k}^{\infty} a_n) = 0. \qquad (12.12) $$

## 12.3. Hyper Infinite Series of Nonnegative Terms.

The theory of series $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n$ with terms that are nonnegative for sufficiently large $n \in \mathbb{N}^\# \setminus \mathbb{N}$ is simpler than the general theory, since such a series either #-converges to a finite or hyperfinite #-limit or diverges to $\infty^\#$, as the next theorem shows.

**Theorem 12.10.** If $a_n \ge 0$ for $n \ge k$, then $\mathrm{Ext}-\sum_{n=1}^{\infty} a_n$ #-converges if its partial sums are bounded or hyper bounded, or #-diverges to $\infty^\#$ if they are not. These are the only possibilities and, in either case, $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n = \{A_n|n \ge k\}$, where $A_n = \mathrm{Ext}-\sum_{i=k}^{n} a_i$.

**Theorem 12.11.** (The Comparison Test) Suppose that

$$ 0 \le a_n \le b_n, n \ge k. \qquad (12.13) $$

Then

$$ (a) \mathrm{Ext}-\sum_{n=k}^{\infty} a_n < \infty^\# \text{ if } \mathrm{Ext}-\sum_{n=k}^{\infty} b_n < \infty^\#. \quad (b) \mathrm{Ext}-\sum_{n=k}^{\infty} a_n = \infty^\# \text{ if } \mathrm{Ext}-\sum_{n=k}^{\infty} b_n = \infty^\# $$

**Theorem 12.12.** (The Integral Test) Let

$$ c_n = f(n), n \ge k, \qquad (12.14) $$

where f is positive; nonincreasing; and locally #-integrable on $[k, \infty^\#)$. Then
$$ \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \qquad (12.15) $$

if and only if

$$ \mathrm{Ext}-\int_{k}^{\infty^{\#}} f(x)d^{\#}x < \infty^{\#}. \qquad (12.16) $$

**Example 12.2.** The integral test implies that the hyper infinite series $\mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} n^{-p}$ converge if $p > 1$ and diverge if $0 < p \le 1$, because the same is true of the integral $\mathrm{Ext}-\int_{a}^{\infty^{\#}} x^{-p}d^{\#}x, a > 1$.

The next theorem is often applicable where the integral test is not.
**Theorem 12.13.** Suppose that $a_n \ge 0$ and $b_n > 0$ for $n \ge k$. Then

$$ (a) \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \text{ if } \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} b_n < \infty^{\#} \text{ and } \overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_n}{b_n} < \infty^{\#}. $$

$$ (b) \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#} \text{ if } \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} b_n = \infty^{\#} \text{ and } \underline{\#-\lim}_{n \to \infty^{\#}} \frac{a_n}{b_n} > 0. $$

**Corollary 12.3.** Suppose that $a_n \ge 0$ and $b_n > 0$ for $n \ge k$, and $\overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_n}{b_n} = L$.

where $0 < L < \infty^{\#}$. Then $\mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n$ and $\mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} b_n$ #-converge or #-diverge together.

**Theorem 12.14.** Suppose that $a_n > 0, b_n > 0$, and

$$ \frac{a_{n+1}}{a_n} \le \frac{b_{n+1}}{b_n}. \qquad (12.17) $$

$$ \text{Then (a) } \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \text{ if } \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} b_n < \infty^{\#}. \text{ (b) } \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#} \text{ if } \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} b_n = \infty^{\#}. $$

**Theorem 12.15.** (The Ratio Test) Suppose that $a_n > 0$ for $n \ge k$. Then

$$ (a) \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \text{ if } \overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} < 1. \quad (b) \mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#} \text{ if } \underline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} > 1. \text{ If } \\ \underline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} \le 1 \le \overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} \qquad (12.18) $$

then the test is inconclusive; that is, $\mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n$ may #-converge or #-diverge.

**Proof.** (a) If $\overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} < 1$, there is a number $r$ such that $0 < r < 1$ and $\frac{a_{n+1}}{a_n} < r$ for $n \in \mathbb{N}^{\#}$ sufficiently large. This can be rewritten as $\frac{a_{n+1}}{a_n} < \frac{r^{n+1}}{r^n}$
Since $\mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} r^n < \infty^{\#}$ Theorem 12.14 (a) with $b_n = r^n$ implies that $\mathrm{Ext}-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#}$.

(b) If $\underline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} > 1$, there is a number $r$ such that $r > 1$ and $\frac{a_{n+1}}{a_n} > r$ for $n \in \mathbb{N}^{\#}$ sufficiently large. This can be rewritten as $\frac{a_{n+1}}{a_n} > \frac{r^{n+1}}{r^n}$. Since
$Ext-\sum_{n=k}^{\infty^{\#}} r^n = \infty^{\#}$Theorem 12.14 (b) with $b_n = r^n$ implies that $Ext-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#}$.

To see that no conclusion can be drawn if (12.18) holds, consider hyper infinite series

$$Ext-\sum_{n=k}^{\infty^{\#}} a_n = Ext-\sum_{n=k}^{\infty^{\#}} n^{-p}. \qquad (12.19)$$

This series #-converges if $p > 1$ or #-diverges if $p \le 1$, however,

$$\overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} = \overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} = 1. \qquad (12.20)$$

**Corollary 12.4.** Suppose that $a_n > 0$ for $n \ge k$ and $\overline{\#-\lim}_{n \to \infty^{\#}} \frac{a_{n+1}}{a_n} = L$. Then

$$(a) \ Ext-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \text{ if } L < 1. (b) \ Ext-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#} \text{ if } L > 1.$$

The test is inconclusive if $L = 1$.

**Theorem 12.16.** (Generalized Raabe's Test) Suppose that $a_n > 0$ for large $n \in \mathbb{N}^{\#} \setminus \mathbb{N}$.

Let $M = \overline{\#-\lim}_{n \to \infty^{\#}} (\frac{a_{n+1}}{a_n} - 1)$ and $m = \overline{\#-\lim}_{n \to \infty^{\#}} (\frac{a_{n+1}}{a_n} - 1)$. Then

$$(a) \ Ext-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \text{ if } M < -1. (b) \ Ext-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#} \text{ if } m > -1.$$

The test is inconclusive if $m \le -1 \le M$.

**Theorem 12.17.** (Generalized Cauchy's Root Test) Suppose that $a_n \ge 0$ for $n \ge k \in \mathbb{N}^{\#} \setminus \mathbb{N}$, then

$$(a) \ Ext-\sum_{n=k}^{\infty^{\#}} a_n < \infty^{\#} \text{ if } \overline{\#-\lim}_{n \to \infty^{\#}} \sqrt[3]{a_n} < 1. (b) \ Ext-\sum_{n=k}^{\infty^{\#}} a_n = \infty^{\#} \text{ if } \overline{\#-\lim}_{n \to \infty^{\#}} \sqrt[3]{a_n} > 1.$$

The test is inconclusive if $\overline{\#-\lim}_{n \to \infty^{\#}} \sqrt[3]{a_n} = 1$.

## 12.4. Absolute and Conditional #-Convergence.

**Definition 12.6.** A series $Ext-\sum_{n=k}^{\infty^{\#}} a_n$ #-converges absolutely, or is absolutely

$$\#-\text{convergent if } Ext-\sum_{n=k}^{\infty^{\#}} |a_n| < \infty^{\#}.$$

**Theorem 12.18.** If $Ext-\sum_{n=k}^{\infty^{\#}} a_n$ #-converges absolutely; then $Ext-\sum_{n=k}^{\infty^{\#}} a_n$ #-converges.

**Theorem 12.19.** (Dirichlet's Test for Hyper Infinite Series) The hyper infinite series

$$Ext-\sum_{n=k}^{\infty^{\#}} a_n b_n \text{ is #-converges if the following conditions are satisfied}$$

(i) $\overline{\#-\lim}_{n \to \infty^{\#}} a_n = 0$,

(ii)

$$Ext-\sum_{n=k}^{\infty^{\#}} |a_{n+1} - a_n| < \infty^{\#} \qquad (12.21)$$

and
(iii) for all $n \ge k$

$$ \mathrm{Ext}-\sum_{i=k}^{n} b_n \le M \qquad (12.22) $$

for some constant $M$.

**Proof.** Let $B_n, n \ge k$ be the partial sum

$$ B_n = \mathrm{Ext}-\sum_{i=k}^{n} b_n \qquad (12.23) $$

Let us consider the partial sums $S_n, n \ge k$ of $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n b_n$, where

$$ S_n = \mathrm{Ext}-\sum_{i=k}^{n} a_n b_n \qquad (12.24) $$

By substituting $b_k = B_k$ and $b_n = B_n - B_{n-1}, n \ge k+1$, into (12.24), we obtain

$$ S_n = a_k b_k + \mathrm{Ext}-\sum_{i=k+1}^{n} a_i (B_i - B_{i-1}), \qquad (12.25) $$

which we rewrite as

$$ S_n = a_n B_n + \mathrm{Ext}-\sum_{i=k}^{n-1} (a_i - a_{i+1}) B_i. \qquad (12.26) $$

Now (12.26) can be viewed as

$$ S_n = T_{n-1} + a_n B_n, \qquad (12.27) $$

where $T_{n-1} = \mathrm{Ext}-\sum_{i=k}^{n-1} (a_i - a_{i+1}) B_i$; that is, $\{T_n\}$ is the hyper infinite sequence of partial sums of the hyper infinite series

$$ \mathrm{Ext}-\sum_{i=k}^{\infty} (a_i - a_{i+1}) B_i. \qquad (12.28) $$

Since $|(a_i - a_{i+1})B_i| \le M|a_i - a_{i+1}|$ from (12.22), the comparison test and (12.21) imply that the series (12.28) #-converges absolutely. Theorem 12.18 now implies that $\{T_n\}_{n \in \mathbb{N}}$ #-converges. Let $T = \#\lim_{n \to \infty} T_n$. Since $B_n$ is bounded (hyperbounded) and $\#\lim_{n \to \infty} a_n = 0$, we infer from (12.27) that

$$ \#\lim_{n \to \infty} S_n = \#\lim_{n \to \infty} T_{n-1} + \#\lim_{n \to \infty} a_n B_n = T. \qquad (12.29) $$

Therefore, $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n b_n$ is #-converges.

**Corollary 12.4. (Abel's Test for Hyper Infinite Series)** The series $\mathrm{Ext}-\sum_{n=k}^{\infty} a_n b_n$ #-converges if $a_{n+1} \le a_n$ for $n \ge k$, $\#\lim_{n \to \infty} a_n = 0$ and $\mathrm{Ext}-\sum_{i=k}^{n} b_n \le M$, for some constant $M$.
**Corollary 12.5.** (Alternating Hyper Infinite Series Test) The series $Ext-\sum_{n=0}^{\infty}(-1)^n a_n$ #-converges if $0 \le a_{n+1} \le a_n$ and #-lim$_{n \to \infty}$ $a_n = 0$.

**Proof.** Let $b_n = (-1)^n$, then $\{|B_n|\}_{n \in \mathbb{N}^\#}$ is a hyper infinite sequence of zeros and ones and therefore bounded. The conclusion now follows from Abel's test.

## 12.5. Grouping Terms in a Hyper Infinite Series.

The terms of a hyper finite sum can be grouped arbitrarily by it hyper finite (but not by countable set of it finite subsets) subsets by inserting corresponding parentheses, see Appendix C. According to the next theorem, the same is true of an hyper infinite series that #-converges or #-diverges to $\pm\infty$.

**Theorem 12.20.** Suppose that $Ext-\sum_{n=k}^{\infty} a_n = A$, where $-\infty \le A \le \infty$. Let $\{n_j\}_{n \in \mathbb{N}^\#}$ be an increasing hyper infinite sequence of integers, with $n_1 \ge k$. Define

$$ 
\begin{align} 
b_1 &= Ext-\sum_{n=k}^{n_1} a_n, \notag \\ 
b_2 &= Ext-\sum_{n=n_1+1}^{n_2} a_n, \tag{12.30} \\ 
\multicolumn{2}{c}{\dots\dots\dots\dots\dots} \notag \\ 
b_r &= Ext-\sum_{n=n_{r-1}+1}^{n_r} a_n \notag 
\end{align} 
$$

Then

$$ Ext-\sum_{j=1}^{\infty} b_{n_j} = A. \tag{12.31} $$

## 12.6. Rearrangement of hyper infinite series.

A hyperfinite sum is not changed by rearranging its terms, see Appendix C. According to the next theorem, we see that every rearrangement of an absolutely #-convergent hyper infinite series has the same sum, but that conditionally #-convergent series fail, spectacularly, to have this property.

**Theorem 12.21.** If $Ext-\sum_{n=1}^{\infty} b_n$ is a rearrangement of an absolutely #-convergent series $Ext-\sum_{n=1}^{\infty} a_n$ then $Ext-\sum_{n=1}^{\infty} b_n$ also #-converges absolutely, and to the same sum.

**Theorem 12.22.** If $\{a_{n_i}\}_{i \in \mathbb{N}^\#}$ and $\{a_{m_j}\}_{j \in \mathbb{N}^\#}$ are respectively the subsequences of all positive and negative terms in a conditionally #-convergent series $Ext-\sum_{n=1}^{\infty} a_n$ then
$$ \mathrm{Ext}-\sum_{i=1}^{\infty^{\#}} a_{n_i} = \infty^{\#} \text{ and } \mathrm{Ext}-\sum_{j=1}^{\infty^{\#}} a_{m_j} = -\infty^{\#}. \qquad (12.32) $$

**Theorem 12.23.** Suppose that $\mathrm{Ext}-\sum_{n=1}^{\infty^{\#}} a_n$ is conditionally #-convergent and $\mu$ and $\nu$ are arbitrarily given in the extended hyperreals; with $\mu \le \nu$. Then the terms of

$\mathrm{Ext}-\sum_{n=1}^{\infty^{\#}} a_n$ can be rearranged to form a series $\mathrm{Ext}-\sum_{n=1}^{\infty^{\#}} b_n$ with partial sums

$B_n = \mathrm{Ext}-\sum_{i=1}^{n} b_i$ such that

$$ \varlimsup_{n \to \infty} B_n = \nu \text{ and } \varliminf_{n \to \infty} B_n = \mu. \qquad (12.33) $$

## 12.7. Multiplication of hyper infinite Series.

Given two hyper infinite series $\mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} a_n$ and $\mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} b_n$ we can arrange all possible products $a_ib_j$, $i,j \ge 0$ in a two-dimensional array:

$$ \begin{array}{ccccccc} a_0 b_0 & a_0 b_1 & a_0 b_2 & a_0 b_3 & \cdots & & \\ a_1 b_0 & a_1 b_1 & a_1 b_2 & a_1 b_3 & \cdots & & \\ a_2 b_0 & a_2 b_1 & a_2 b_2 & a_2 b_3 & \cdots & & \\ a_3 b_0 & a_3 b_1 & a_3 b_2 & a_3 b_3 & \cdots & & \\ \cdot & \cdot & \cdot & \cdot & \cdots & & \end{array} \qquad (12.34) $$

where the subscript on a is constant in each row and the subscript on b is constant in each column. Any sensible definition of the product

$$ \left( \mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} a_n \right) \left( \mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} b_n \right) \qquad (12.35) $$

clearly must involve every product in this array exactly once; thus, we might define the product of the two series to be the series $\mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} p_n$, where $\{p_n\}_{n \in \mathbb{N}^{\#}}$ is a hyper infinite sequence obtained by ordering the products in (12.34) according to some method that chooses every product exactly once.

**Theorem 12.24.** Let $\mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} a_n = A$ and $\mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} b_n = B$, where A and B are finite or hyperfinite, and at least one term of each series is nonzero. Then $\mathrm{Ext}-\sum_{n=0}^{\infty^{\#}} p_n = A \times B$ for every hyper infinite sequence $\{p_n\}_{n \in \mathbb{N}^{\#}}$ obtained by ordering the products in
(12.34) if and only if $Ext-\sum_{n=0}^{\infty} a_n$ and $Ext-\sum_{n=0}^{\infty} b_n$ #-converge absolutely:

Moreover, in this case, $Ext-\sum_{n=0}^{\infty} p_n$ #-converges absolutely.

**Definition 12.7.** The Cauchy product of $Ext-\sum_{n=0}^{\infty} a_n$ and $Ext-\sum_{n=0}^{\infty} b_n$ is $Ext-\sum_{n=0}^{\infty} c_n$, where

$$c_n = Ext-\sum_{j=0}^{n} a_j b_{n-j}. \qquad (12.36)$$

Thus, $c_n$ is the external sum of all products $a_l b_k$, where $i \ge 0, j \ge 0$, and $i + j = n$; thus,

$$c_n = Ext-\sum_{j=0}^{n} a_j b_{n-j} = Ext-\sum_{j=0}^{n} b_j a_{n-j}. \qquad (12.37)$$

**Theorem 12.25.** If $Ext-\sum_{n=0}^{\infty} a_n$ and $Ext-\sum_{n=0}^{\infty} b_n$ #-converge absolutely to sums A and B, then the Cauchy product $Ext-\sum_{j=0}^{n} a_j b_{n-j}$ #-converges absolutely to AB.

**Theorem 12.26.** Let $f(\alpha) = Ext-\sum_{n=0}^{\infty} \frac{\alpha^n}{n!}$ and $f(\beta) = Ext-\sum_{n=0}^{\infty} \frac{\beta^n}{n!}$, then

$$f(\alpha)f(\beta) = f(\alpha + \beta). \qquad (12.38)$$

**Proof.** From Eq. (12.37) we obtain

$$c_n = Ext-\sum_{n=0}^{m} \frac{\alpha^{n-m} \beta^m}{(n-m)!m!} = \frac{1}{n!} \left( Ext-\sum_{n=0}^{m} \binom{n}{m} \alpha^{n-m} \beta^m \right) = Ext-\sum_{n=0}^{\infty} \frac{(\alpha + \beta)^n}{n!} \qquad (12.39)$$

Thus

$$f(\alpha)f(\beta) = Ext-\sum_{n=0}^{\infty} \frac{(\alpha + \beta)^n}{n!} = f(\alpha + \beta). \qquad (12.40)$$

## 12.8. Double Hyper Infinite Sequences.

**Definition 12.8.** A double hyper infinite sequence of hyperreal numbers $\mathbb{R}_c^\#$ (complex numbers $\mathbb{C}_c^\# = \mathbb{R}_c^\# + i\mathbb{R}_c^\#$) is a $\mathbb{R}_c^\#$-valued ($\mathbb{C}_c^\#$-valued) function $s: \mathbb{N}^\# \times \mathbb{N}^\# \to \mathbb{R}_c^\#$ or $s: \mathbb{N}^\# \times \mathbb{N}^\# \to \mathbb{C}_c^\#$. We shall use the notation $\{s_{n,m}\}_{n,m \in \mathbb{N}^\#}$ or simply $s_{nm}$.

**Definition 12.9.** We say that a double hyper infinite sequence $s_{n,m}$ #-converges to $a \in \mathbb{C}_c^\#$ and we write $\#-\lim_{n,m \to \infty} s_{n,m} = a$, if the following condition is satisfied: for every $\varepsilon > 0$, $\varepsilon \approx 0$, there exists $N \in \mathbb{N}^\#$ such that $|s_{n,m} - a| < \varepsilon$ if $n, m \ge N$.

**Theorem 12.27.** (Uniqueness of Double #-Limits). A double hyper infinite $\mathbb{C}_c^\#$-valued sequence has at most one #-limit.

**Definition 12.10.** A double hyper infinite sequence $s_{n,m}$ is called bounded (hyper bounded) if there exists finite (hyperfinite) number $M \in \mathbb{R}_c^\#$, $M > 0$ such that $|s_{n,m}| \le M, \forall n, m \in \mathbb{N}^\#$.
**Theorem12.28.** A #-convergent double $C_c^#$-valued hyper infinite sequence is bounded or hyper bounded.

**Definition 12.11.** A double $C_c^#$-valued hyper infinite sequence $s_{n,m}$ is called a Cauchy sequence if and only if for every $\varepsilon > 0, \varepsilon \approx 0$, there exists a hypernatural number $N \in \mathbb{N}^\# \setminus \mathbb{N}$ such that $|s_{p,q} - s_{n,m}| < \varepsilon, \forall p(\forall q(\forall n \ge N) \text{ and } \forall q(\forall m \ge N))$.

**Theorem 12.28.** (Cauchy Convergence Criterion for Double hyper infinite Sequences). A double $C_c^#$-valued hyper infinite sequence $s_{n,m}, n, m \in \mathbb{N}^#$ #-converges if and only if it is a Cauchy sequence.

**Definition 12.12.** Let $s_{n,m}$ be a double $\mathbb{R}_c^#$-valued hyper infinite sequence.

(i) If $s_{n,m} \le s_{j,k}, \forall n \forall j \forall m \forall k (n \le j \land m \le k)$, $n,m,j,k \in \mathbb{N}^#$, we say the sequence $s_{n,m}$ is increasing.

(ii) $s_{n,m} \ge s_{j,k}, \forall n \forall j \forall m \forall k (n \le j \land m \le k)$, $n,m,j,k \in \mathbb{N}^#$, we say the sequence $s_{n,m}$ is decreasing.

(ii) If $s_{n,m}$ is either increasing or decreasing, then we say it is monotone.

**Definition 12.13.** For a double sequence $s_{n,m}$, the #-limits

$\#\lim_{n \to \#} \infty^# (\#\lim_{m \to \#} \infty^# s_{n,m})$ and $\#\lim_{m \to \#} \infty^# (\#\lim_{n \to \#} \infty^# s_{n,m})$ are called repeated #-limits.

**Theorem 12.29.** Let $\#\lim_{n,m \to \#} \infty^# s_{n,m} = a$. Then $\#\lim_{m \to \#} \infty^# (\#\lim_{n \to \#} \infty^# s_{n,m}) = a$ if and only if $\#\lim_{n \to \#} \infty^# s_{n,m}$ exists for each $m \in \mathbb{N}^#$.

**Theorem 12.30.** Let $\#\lim_{n,m \to \#} \infty^# s_{n,m} = a$. Then the repeated #-limits $\#\lim_{n \to \#} \infty^# (\#\lim_{m \to \#} \infty^# s_{n,m})$ and $\#\lim_{m \to \#} \infty^# (\#\lim_{n \to \#} \infty^# s_{n,m})$ exist and both are equal to a if and only if (i) $\#\lim_{n \to \#} \infty^# s_{n,m}$ exists for each $m \in \mathbb{N}^#$, and (ii) $\#\lim_{m \to \#} \infty^# s_{n,m}$ exists for each $n \in \mathbb{N}^#$.

**Theorem 12.31.** If $s_{n,m}$ is a double sequence such that the repeated #-limit $\#\lim_{m \to \#} \infty^# (\#\lim_{n \to \#} \infty^# s_{n,m}) = a$ and the #-limit $\#\lim_{n \to \#} \infty^# s_{n,m}$ exists uniformly in $m \in \mathbb{N}^#$, then the double #-limit $\#\lim_{n,m \to \#} \infty^# s_{n,m} = a$.

**Theorem 12.32.** (Monotone Convergence Theorem). A monotone double $\mathbb{R}_c^#$-valued hyper infinite sequence is #-convergent if and only if it is bounded (hyper bounded).

Further: (i) If $s_{n,m}$ is increasing and bounded (hyper bounded) above, then $\#\lim_{m \to \#} \infty^# (\#\lim_{n \to \#} \infty^# s_{n,m}) = \#\lim_{n \to \#} \infty^# (\#\lim_{m \to \#} \infty^# s_{n,m}) = \#\lim_{n,m \to \#} \infty^# s_{n,m}$.

(ii) If $s_{n,m}$ is decreasing and bounded (hyper bounded) below, then $\#\lim_{m \to \#} \infty^# (\#\lim_{n \to \#} \infty^# s_{n,m}) = \#\lim_{n \to \#} \infty^# (\#\lim_{m \to \#} \infty^# s_{n,m}) = \#\lim_{n,m \to \#} \infty^# s_{n,m}$.

**Theorem12.33.** (The Sandwich Theorem). Suppose that $x_{n,m}, s_{n,m}$, and $y_{n,m}$ are double $\mathbb{R}_c^#$-valued hyper infinite sequences such that $x_{n,m} \le s_{n,m} \le y_{n,m}, \forall n,m \in \mathbb{N}^#$, and $\#\lim_{n,m \to \#} \infty^# x_{n,m} = \#\lim_{n,m \to \#} \infty^# y_{n,m}$. Then $s_{n,m}$ is #-convergent and $\#\lim_{n,m \to \#} \infty^# x_{n,m} = \#\lim_{n,m \to \#} \infty^# y_{n,m} = \#\lim_{n,m \to \#} \infty^# s_{n,m}$.

**Definition 12.14.** Let $s_{n,m}$ be a double $C_c^#$-valued hyper infinite sequence and let $(k_1, r_1) < (k_2, r_2) < \dots < (k_n, r_n) < \dots$ be a strictly increasing sequences of pairs of hypernatural numbers. Then the sequence $s_{k_n, r_m}$ is called a subsequence of $s_{n,m}$.

**Theorem12.34.** If a double $C_c^#$-valued hyper infinite sequence $s_{n,m}$ #-converges to number $a \in C_c^#$, then any hyper infinite subsequence of $s_{n,m}$ also #-converges to $a$.

**Theorem12.35.** If the repeated #-limits of a double sequence $s_{n,m}$ exist and
satisfy $\#\lim_{m \to \#} \infty^\# (\#\lim_{n \to \#} \infty^\# s_{n,m}) = \#\lim_{n \to \#} \infty^\# (\#\lim_{m \to \#} \infty^\# s_{n,m}) = a$, then the

repeated $\#\text{limits}$ for any subsequence $s_{p_n, q_m}$ exist and satisfy
$\#\lim_{m \to \#} \infty^\# (\#\lim_{n \to \#} \infty^\# s_{p_n, q_m}) = \#\lim_{n \to \#} \infty^\# (\#\lim_{m \to \#} \infty^\# s_{p_n, q_m}) = a$.

**Theorem12.36.** Every double $\mathbb{R}_c^\#$-valued hyper infinite sequence has a monotone hyper infinite subsequence.

**Theorem12.37.** (Bolzano-Weierstrass Theorem). A bounded (hyper bounded) double $\mathbb{R}_c^\#$-valued hyper infinite sequence has a $\#$-convergent monotone subsequence.

## 12.9. External Double Hyper Infinite Series.

**Definition 12.15.** Let $z: \mathbb{N}^\# \times \mathbb{N}^\# \to \mathbb{C}_c^\#$ be external hyper infinite double sequence of complex numbers $\mathbb{C}_c^\#$$\in$ let $s_{n,m}$ be the double hyper infinite sequence defined by the equation

$$s_{n,m} = \mathrm{Ext} - \sum_{i=1}^{n} \left( \mathrm{Ext} - \sum_{j=1}^{m} z_{i,j} \right). \qquad (12.41)$$

The pair $(z, s)$ is called a double hyper infinite series and is denoted by the symbol

$$\mathrm{Ext} - \sum_{n=1, m=1}^{\infty} z_{n,m} \qquad (12.42)$$

or, more briefly by $\mathrm{Ext} - \sum_{n,m=1}^{\infty} z_{n,m}$. Each number $z_{n,m}$ is called a term of the double series and each $s_{n,m}$ is called a partial sum.

**Definition 12.16.** We say that the double series $\mathrm{Ext} - \sum_{n,m=1}^{\infty} z_{n,m}$ is $\#\text{-convergent}$ to the sum $s$ if $\#\lim_{n,m \to \#} \infty^\# s_{n,m} = s$. If no such $\#\text{-limit}$ exists, we say that the double series $\mathrm{Ext} - \sum_{n,m=1}^{\infty} z_{n,m}$ is $\#\text{-divergent}$.

**Definition 12.17.** The hyper infinite series

$$\mathrm{Ext} - \sum_{n=1}^{\infty} \left( \mathrm{Ext} - \sum_{m=1}^{\infty} z_{n,m} \right) \qquad (12.43)$$

and

$$\mathrm{Ext} - \sum_{m=1}^{\infty} \left( \mathrm{Ext} - \sum_{n=1}^{\infty} z_{n,m} \right) \qquad (12.44)$$

are called repeated hyper infinite series.

**Theorem 12.38.** If the double hyper infinite series $\mathrm{Ext} - \sum_{n=1, m=1}^{\infty} z_{n,m}$ is $\#\text{-convergent}$, then

$$\#\lim_{n,m \to \infty} z_{n,m} = 0. \qquad (12.45)$$

**Theorem 12.39.** (Cauchy $\#\text{-Convergence Criterion for Double hyper infinite Series}$.)

A double hyper infinite series $\mathrm{Ext} - \sum_{n=1, m=1}^{\infty} z_{n,m}$ $\#\text{-converges}$ if and only if its sequence of
partial sums $s_{n,m}$ is Cauchy.

**Theorem12.40.** If the double series $Ext-\sum_{n=1,m=1}^{\infty} z_{n,m}$ #-converges to $s_1$ and $Ext-\sum_{n=1,m=1}^{\infty} u_{n,m}$ #-converges to $s_2$, then: (i) $Ext-\sum_{n=1,m=1}^{\infty} z_{n,m} + Ext-\sum_{n=1,m=1}^{\infty} u_{n,m} = s_1 + s_2$.

(ii) $Ext-\sum_{n=1,m=1}^{\infty} c \times z_{n,m} = c \times (Ext-\sum_{n=1,m=1}^{\infty} z_{n,m})$.

**Theorem12.41.** Suppose that the double series $Ext-\sum_{n=1,m=1}^{\infty} z_{n,m}$ is #-convergent, with sum $s$. Then the repeated series $Ext-\sum_{n=1}^{\infty} (Ext-\sum_{m=1}^{\infty} z_{n,m})$ and

$Ext-\sum_{m=1}^{\infty} (Ext-\sum_{n=1}^{\infty} z_{n,m})$ are both #-convergent with sum $s$ if and only if for every

$m \in \mathbb{N}^{\#}$, the series $Ext-\sum_{n=1}^{\infty} z_{n,m}$ is #-convergent, and for every $n \in \mathbb{N}^{\#}$, the

series $Ext-\sum_{m=1}^{\infty} z_{n,m}$ is #-convergent.

## 12.10. Interchanging the order of summation of hyper infinite sum.

**Theorem 12.**. Assume that

$$ Ext-\sum_{i=1}^{\infty} \left( Ext-\sum_{k=1}^{\infty} |a_{jk}| \right) < \infty^{\#}. \qquad (12.46) $$

Then

$$ Ext-\sum_{i=1}^{\infty} \left( Ext-\sum_{k=1}^{\infty} |a_{jk}| \right) = Ext-\sum_{k=1}^{\infty} \left( Ext-\sum_{j=1}^{\infty} |a_{jk}| \right) \qquad (12.47) $$

## 13. Hyper infinite sequences and series of $\mathbb{R}^c$-valued functions.

### 13.1. Uniform #-Convergence

If $f_k, f_{k+1}, \dots, f_n, \dots, n \in \mathbb{N}^{\#}$ are $\mathbb{R}^c$-valued functions defined on a subset $D \subset \mathbb{R}^c$ of the hyperreals, we say that $\{f_n\}_{n \in \mathbb{N}^{\#}}$ is a hyper infinite sequence of functions on $D$. If the sequence of values $\{f_n(x)\}_{n \in \mathbb{N}^{\#}}$ #-converges for each $x$ in some subset $S$ of $D$, then $\{f_n\}_{n \in \mathbb{N}^{\#}}$ defines a #-limit function on $S$. The formal definition is as follows.

**Definition 13.1.** Suppose that $\{f_n\}_{n \in \mathbb{N}^{\#}}$ is a hyper infinite sequence of functions on $D \subset \mathbb{R}^c$ and the hyper infinite sequence of values $\{f_n(x)\}_{n \in \mathbb{N}^{\#}}$ #-converges for each $x$ in some subset $S$ of $D$. Then we say that $\{f_n\}_{n \in \mathbb{N}^{\#}}$ #-converges pointwise on $S$ to the
\#-limit function $f$, defined by

$$f(x) = \#\lim_{n \to \infty} f_n(x), x \in S. \qquad (13.1)$$

**Definition 13.2.** Let $f$ be a function defined on $S \subset \mathbb{R}_c^\#$ and there exist $\sup_{x \in S}|f(x)|$, then we set

$$\|f\|_S = \sup_{x \in S}|f(x)|. \qquad (13.2)$$

**Lemma 13.1.** If $g$ and $h$ are defined on $S$, then $\|g+h\|_S \le \|g\|_S + \|h\|_S$ and $\|g \times h\|_S \le \|g\|_S \times \|h\|_S$. Moroever if either $g$ or $h$ is bounded on $S$, then $\|g-h\|_S \ge \|g\|_S - \|h\|_S$.

**Definition 13.2.** A hyper infinite sequence $\{f_n\}_{n \in \mathbb{N}^\#}$ of functions defined on a set $S$ \#-converges uniformly to the \#-limit function $f$ on $S$ if $\#\lim_{n \to \infty} \|f_n - f\|_S = 0$. Thus, $f_n$ \#-converges uniformly to $f$ on $S$ if for each $\varepsilon > 0$, $\varepsilon \approx 0$, there is an integer $N \in \mathbb{N}^\# \setminus \mathbb{N}$ such that

$$\|f_n - f\| < \varepsilon \text{ if } n \ge N. \qquad (13.3)$$

**Theorem 13.1.** Let $f_n, n \in \mathbb{N}^\#$ be hyper infinite sequence defined on $S$. Then (a) $f_n$ \#-converges pointwise to $f$ on $S$ if and only if there is, for each $\varepsilon > 0$, $\varepsilon \approx 0$, and $x \in S$, an integer $N \in \mathbb{N}^\# \setminus \mathbb{N}$ which may depend on $x$ as well as $\varepsilon$ such that $|f_n(x) - f(x)| < \varepsilon$ if $n \ge N$; (b) $f_n$ \#-converges uniformly to $f$ on $S$ if and only if there is for each $\varepsilon > 0$, $\varepsilon \approx 0$, an integer $N \in \mathbb{N}^\# \setminus \mathbb{N}$ which depends only on and not on any particular $x$ in $S$ such that $|f_n(x) - f(x)| < \varepsilon$ for all $x \in S$ if $n \ge N$.

**Theorem 13.2.** If $f_n$ \#-converges uniformly to $f$ on $S$, then $f_n$ \#-converges pointwise to $f$ on $S$. The converse is false; that is pointwise \#-convergence does not imply uniform \#-convergence.

**Theorem 13.3.** (Cauchy's Uniform \#-Convergence Criterion) A sequence of functions $f_n$ \#-converges uniformly on a set $S$ if and only if for each $\varepsilon > 0$, $\varepsilon \approx 0$, there is an integer $N \in \mathbb{N}^\# \setminus \mathbb{N}$ such that

$$\|f_n - f_m\|_S < \varepsilon \text{ if } n, m \ge N. \qquad (13.4)$$

**Theorem 13.4.** If $f_n$ \#-converges uniformly to $f$ on $S$ and each $f_n$ is \#-continuous at a point $x_0 \in S$; then so is $f$. Similar statements hold for \#-continuity from the right and left.

**Theorem 13.5.** Suppose that $f_n$ \#-converges uniformly to $f$ on $S = [a, b]$. Assume that $f$ and all $f_n$ are \#-integrable on $[a, b]$. Then

$$\mathrm{Ext}- \int_a^b f(x) d^\# x = \#\lim_{n \to \infty} \left( \mathrm{Ext}- \int_a^b f_n(x) d^\# x \right). \qquad (13.5)$$

**Proof.** Since

$$\left| \mathrm{Ext}- \int_a^b f(x) d^\# x - \mathrm{Ext}- \int_a^b f_n(x) d^\# x \right| \le \mathrm{Ext}- \int_a^b |f(x) - f_n(x)| d^\# x \le (b-a) \|f - f_n\|_S \qquad (13.6)$$

and $\#\lim_{n \to \infty} \|f - f_n\|_S = 0$, the Eq. (13.5) follows.
**Theorem 13.6.** Suppose that $f_n(x)$ #-converges pointwise to $f$ and each $f_n(x)$ is #-integrable on $[a, b]$. Then

(a) If the #-convergence is uniform, then $f(x)$ is #-integrable on $[a, b]$ and (13.5) holds.

(b) If the sequence $\|f_n\|_{[a,b]}$ is bounded and $f(x)$ is #-integrable on $[a, b]$, then (13.5) holds.

**Theorem 13.7.** Suppose that $f_n^{\prime\#}(x)$ is #-continuous on $[a, b]$ for all $n \in \mathbb{N}^\#$ and $\{f_n^{\prime\#}\}_{n \in \mathbb{N}^\#}$ #-converges uniformly on $[a, b]$. Suppose also that $\{f_n(x_0)\}_{n \in \mathbb{N}^\#}$

#-converges for some $x_0 \in [a, b]$. Then $\{f_n(x)\}_{n \in \mathbb{N}^\#}$ #-converges uniformly on $[a, b]$ to

a #-differentiable #-limit function $f(x)$ and

$$f^{\prime\#}(x) = \#\lim_{n\to\infty\#} f_n^{\prime\#}(x), x \in (a, b), \qquad (13.7)$$

while

$$f_{+}^{\prime\#}(a) = \#\lim_{n\to\infty\#} f_{n}^{\prime\#}(a+), f_{-}^{\prime\#}(b) = \#\lim_{n\to\infty\#} f_{n}^{\prime\#}(b-). \qquad (13.8)$$

## 13.2. Hyper Infinite Series of Functions.

**Definition 13.3.** If $\{f_j(x)\}_{j=k}^{\infty}$ is a hyper infinite sequence of $\mathbb{R}_c^\#$-valued functions defined on a set $D \subset \mathbb{R}_c^\#$ of hyperreals, then

$$\mathrm{Ext}-\sum_{j=k}^{\infty} f_j(x) \qquad (13.9)$$

is an hyper infinite series of functions on $D$. The partial sums of, $\mathrm{Ext}-\sum_{j=k}^{\infty} f_j(x)$ are defined by

$$F_n(x) = \mathrm{Ext}-\sum_{j=k}^{n} f_j(x), n \in \mathbb{N}^\#. \qquad (13.10)$$

If $F_n(x)$ #-converges pointwise to a function $F$ on a subset $S \subset D$, we say that $\mathrm{Ext}-\sum_{j=k}^{n} f_j(x)$ #-converges pointwise to the sum $F(x)$ on $S$, and write

$$F(x) = \mathrm{Ext}-\sum_{j=k}^{\infty} f_j(x). \qquad (13.11)$$

If $F_n(x)$ #-converges uniformly to $F(x)$ on $S$, we say that $\mathrm{Ext}-\sum_{j=k}^{n} f_j(x)$ #-converges uniformly to $F(x)$ on $S$.

**Example 13.1.** The functions $f_j(x) = x^j, j \in \mathbb{N}^\#$ define the hyper infinite series

$\mathrm{Ext}-\sum_{j=0}^{\infty} x^j$ on $D = (-\infty^\#, \infty^\#)$. The n-th partial sum of the series is $F_n(x) = \mathrm{Ext}-\sum_{j=0}^{n} x^j$, or, in closed form,

$$F_n(x) = \begin{cases} \frac{1-x^n}{1-x} & x \neq 1 \\ n+1 & x=1 \end{cases} \qquad (13.12)$$

Therefore $\{F_n\}$ #-converges pointwise to if $|x| < 1$ and #-diverges if $|x| > 1$, hence,
we get $F(x) = \text{Ext}-\sum_{j=0}^{\infty} x^j = (1-x)^{-1}, -1 < x < 1$. Since the difference $F(x) - F_n(x) =$
$= \frac{x^n}{1-x}$ can be made arbitrarily infinite large by taking $x$ infinite close to 1,
$\|F - F_n\|_{(-1,1)} = \infty^\#$ so the #-convergence is not uniform on $(-1,1)$. Neither is it
uniform on any interval $(-1, r]$ with $1 < r < 1$, since $\|F - F_n\|_{[-r,r]} = r^n/(1-r)$ and
\#-lim$_{n \to \infty^\#}$ $r^n = \infty^\#$. Put another way, the series \#-converges uniformly on \#-closed
subsets of $(-1,1)$.

**Theorem 13.8.** (Cauchy's Uniform #-Convergence Criterion) A hyper infinite series
$\text{Ext}-\sum_{i=0}^{\infty} f_i(x)$ \#-converges uniformly on a set $S \subset \mathbb{R}_c^\#$ if and only if for each $\varepsilon > 0, \varepsilon \approx 0$
there is an hyperinteger $N \in \mathbb{N}^\#$ such that

$$
\left\| \operatorname{Ext} - \sum_{n}^{m} f_{i}(x) \right\|_{S} < \varepsilon \qquad (13.13)
$$

if $m \ge n \ge N$.

**Corollary 13.1.** If $\text{Ext}-\sum_{i=0}^{\infty} f_i(x)$ \#-converges uniformly on $S$, then $\#\text{-lim}_{n\to\infty^\#} \|f_n\|_S = 0$.

**Theorem 13.9.** (Weierstrass's Test) The hyper infinite series $\text{Ext}-\sum_{i=0}^{\infty} f_i(x)$ \#-converges uniformly on $S$ if

$$
\|f_n\|_S \le M_n, n \ge k, \qquad (13.14)
$$

where $\text{Ext}-\sum_{n=k}^{\infty} M_n < \infty^\#$.

**Theorem 13.10.** (Dirichlet's Test for Uniform #-Convergence) The hyper infinite vseries
$\text{Ext}-\sum_{n=k}^{\infty} f_n(x)g_n(x)$ \#-converges uniformly on $S$ if $f_n$ \#-converges uniformly to zero on $S$,
$\text{Ext}-\sum_{n=k}^{\infty} (f_{n+1}(x)-f_n(x))$ \#-converges absolutely uniformly on $S$, and

$$
\left\| \operatorname{Ext} - \sum_{i=k}^{n} g_i(x) \right\|_S \le M, \qquad (13.14)
$$

where $n \ge k$, for some constant $M$.

**Corollary 13.2.** The hyper infinite series $\text{Ext}-\sum_{n=k}^{\infty} f_n(x)g_n(x)$ \#-converges uniformly on $S$ if $f_{n+1}(x) \le f_n(x)$, $x \in S, n \ge k$, $\{f_n\}$ \#-converges uniformly to zero on $S$, and

$$
\left\| \text{Ext} - \sum_{i=k}^{n} g_i(x) \right\|_S \le M, \qquad (13.15)
$$

for some constant $M$.

13.3.#-Continuity, #-Differentiability, and Integrability of
hyper infinite Series.

**Theorem 13.11.** If $\text{Ext}-\sum_{n=k}^{\infty} f_n(x)$ \#-converges uniformly to $F(x)$ on $S$ and each $f_n$ is
#-continuous at a point $x_0$ in S, then so is $F(x)$. Similar statements hold for #-continuity from the right and left.

**Theorem 13.12.** Suppose that $Ext-\sum_{n=k}^{\infty}f_n(x)$ #-converges uniformly to $F(x)$ on $S = [a, b]$ Assume that $F(x)$ and $f_n(x)$, $n \ge k$, are integrable on $[a, b]$. Then

$$ Ext-\int_{a}^{b} F(x)d^{\#}x = Ext-\sum_{n=k}^{\infty}\left( Ext-\int_{a}^{b} f_{n}(x)d^{\#}x \right). \qquad (13.16) $$

**Theorem 13.13.** Suppose that $f_n$ is #-continuously #-differentiable on $[a, b]$ for each $n \ge k$, $Ext-\sum_{n=k}^{\infty}f_n(x_0)$ #-converges for some $x_0 \in [a, b]$ and $Ext-\sum_{n=k}^{\infty}f_n^{\#}(x)$ #-converges uniformly on $[a, b]$. $Ext-\sum_{n=k}^{\infty}f_n(x)$ #-converges uniformly on $[a, b]$ to a #-differentiable function $F(x)$, and $F^{\prime\#}(x) = Ext-\sum_{n=k}^{\infty}f_n^{\#}(x)$, $a < x < b$, while $F^{\prime\#}(a+) = Ext-\sum_{n=k}^{\infty}f_n^{\#}(a+)$ and $F^{\prime\#}(b-) = Ext-\sum_{n=k}^{\infty}f_n^{\#}(b-)$.

# 14. Hyper Infinite Power Series.

## 14.1. The convergence properties of hyper infinite power series.

**Definition 14.1.** A hyper infinite series of the form

$$ Ext-\sum_{n=0}^{\infty} a_n(x - x_0)^n \qquad (14.1) $$

where $x_0 \in \mathbb{R}^\#$ and $a_n \in \mathbb{R}^\#$, $n \in \mathbb{N}^\#$ is called a hyper infinite power series in $(x - x_0)$. The following theorem summarizes the #-convergence properties of hyper infinite power series.

**Theorem 14.1.** In connection with the hyper infinite power series (14.1) define $R$ in the extended hyperreals by

$$ \frac{1}{R} = \overline{\#-\lim_{n \to \infty}} \sqrt[n]{|a_n|} \qquad (14.2) $$

In particular, $R = 0$ if $\overline{\#-\lim_{n \to \infty}} \sqrt[n]{|a_n|} = \infty^\#$, and $R = \infty^\#$ if $\overline{\#-\lim_{n \to \infty}} \sqrt[n]{|a_n|} = 0$. Then the hyper infinite power series #-converges:

(a) only for $x = x_0$ if $R = 0$

(b) for all $x \in \mathbb{R}^\#$ if $R = \infty^\#$, and absolutely uniformly in every bounded set;

(c) for $x \in (x_0 - R, x_0 + R)$ if $0 < R < 1$, and absolutely uniformly in every closed subset of this interval.

The series #-diverges if $|x - x_0| > R$. No general statement can be made concerning #-convergence at the endpoints $x = x_0 + R$ and $x = x_0 + R$: the series may #-converge absolutely or conditionally at both; #-converge conditionally at one and #-diverge at the other; or #-diverge at both.

**Theorem 14.2.** The radius of #-convergence of $Ext-\sum_{n=0}^{\infty}a_n(x - x_0)^n$ is given by
$$ \frac{1}{R} = \#\text{-lim}_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| \qquad (14.3) $$

if the #-limit exists in the extended hyperreals.

**Example 14.1.** For the hyper infinite power series

$$ \operatorname{Ext}-\sum_{n=0}^{\infty} \frac{x^n}{n!} \qquad (14.4) $$

one obtains that

$$ \#\text{-lim}_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = \#\text{-lim}_{n\to\infty} \frac{n!}{(n+1)!} = \#\text{-lim}_{n\to\infty} \frac{1}{n+1} = 0. \qquad (14.4') $$

Therefore, $R = \infty^{\#}$; that is, the series #-converges for all $x \in \mathbb{R}_c^{\#}$, and absolutely uniformly

in every bounded set.

**Theorem 14.3.** A hyper infinite power series

$$ f(x) = \operatorname{Ext}-\sum_{n=0}^{\infty} a_n (x - x_0)^n \qquad (14.5) $$

with positive radius of #-convergence $R$ is #-continuous and #-differentiable in its interval of #-convergence; and its #-derivative can be obtained by #-differentiating term by term; that is;

$$ f'^{\#}(x) = \operatorname{Ext}-\sum_{n=0}^{\infty} na_n(x - x_0)^{n-1} \qquad (14.6) $$

which can also be written as

$$ f'^{\#}(x) = \operatorname{Ext}-\sum_{n=0}^{\infty} (n+1)a_{n+1}(x-x_0)^n \qquad (14.7) $$

This hyper infinite series also has radius of #-convergence $R$.

**Theorem 14.4.** A hyper infinite power series

$$ f(x) = \operatorname{Ext}-\sum_{n=0}^{\infty} a_n (x - x_0)^n \qquad (14.8) $$

with positive radius of #-convergence $R$ has #-derivatives of all orders in its interval of #-convergence, which can be obtained by repeated term by term #-differentiation thus,

$$ \begin{aligned} f^{(n)^\#}(x) &= \operatorname{Ext}-\sum_{n=k}^{\infty} n(n-1)\cdots(n-k+1)a_n (x-x_0)^n = \\ &= \operatorname{Ext}-\sum_{n=k}^{\infty} \left[ \left( \operatorname{Ext}-\prod_{j=n-k+1}^{n} j \right) a_n (x-x_0)^n \right]. \end{aligned} \qquad (14.9) $$

The radius of #-convergence of each of these hyper infinite series is $R$.

**Corollary 14.1.** (Uniqueness of hyper infinite Power Series) If

$$ \operatorname{Ext}-\sum_{n=0}^{\infty} a_n (x - x_0)^n = \operatorname{Ext}-\sum_{n=0}^{\infty} b_n (x - x_0)^n \qquad (14.10) $$

for all $x$ in some interval $(x_0 - r, x_0 + r)$ then

$$ a_n = b_n, n \ge 0. \qquad (14.11) $$

**Corollary 14.2.** If
$$f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} a_n (x - x_0)^n, |x - x_0| < R \quad (14.12)$$

then

$$a_n = \frac{f^{(n)\#}(x)}{n!}. \quad (14.13)$$

**Theorem 14.5.** If $x_1$ and $x_2$ are in the interval of #-convergence of

$$f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} a_n (x - x_0)^n \quad (14.14)$$

Then

$$\operatorname{Ext} - \int_{x_1}^{x_2} f(x) d^\# x = \operatorname{Ext} - \sum_{n=0}^{\infty} \frac{a_n}{n+1} \left[ (x_2 - x_0)^{n+1} - (x_1 - x_0)^{n+1} \right] \quad (14.15)$$

that is, a hyper infinite power series may be integrated term by term between any two points in its interval of #-convergence.

**Theorem 14.6.** Suppose that $f(x)$ is hyper infinitely #-differentiable on an interval $I$ and

$$\#\operatorname{lim}_{n \to \infty} \frac{f^n}{n!} \left\| f^{(n)\#}(x) \right\|_I = 0. \quad (14.16)$$

Then, if $x_0 \in I^0$, the hyper infinite Taylor series

$$\operatorname{Ext} - \sum_{n=0}^{\infty} \frac{f^{(n)\#}(x)}{n!} (x - x_0)^n \quad (14.17)$$

#-converges uniformly to $f(x)$ on $I_r = I \cap [x_0 - r, x_0 + r]$.

**Theorem 14.7.** If

$$f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} a_n (x - x_0)^n, |x - x_0| < R_1 \quad (14.18)$$

and

$$g(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} b_n (x - x_0)^n, |x - x_0| < R_2 \quad (14.19)$$

and $\alpha$ and $\beta$ are constants, then

$$\alpha f(x) + \beta g(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} (\alpha a_n + \beta b_n) (x - x_0)^n, |x - x_0| < R, \quad (14.20)$$

where $R \ge \min\{R_1, R_2\}$.

**Theorem 14.8.** If $f(x)$ and $g(x)$ are given by Eq.(14.19) and Eq.(14.20) correspondingly, then

$$f(x)g(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} c_n (x - x_0)^n, |x - x_0| < R, \quad (14.21)$$

where

$$c_n = \operatorname{Ext} - \sum_{j=0}^{n} a_j b_{n-j} = \sum_{j=0}^{n} a_{n-j} b_j, \quad (14.22)$$

$n \in \mathbb{N}^\#$ and $R \ge \min\{R_1, R_2\}$.

**Theorem 14.9.** (Generalized Abel's Theorem) Let $f(x)$ be defined by a hyper infinite
power series

$$f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} a_n (x - x_0)^n, |x - x_0| < R \quad (14.23)$$

with finite or hyperfinite radius of #-convergence $R \in \mathbb{R}_c^\#$.

(a) If $\operatorname{Ext} - \sum_{n=0}^{\infty} a_n R^n$ #-converges, then

$$\#- \lim_{x \to \# (x_0+R)-} f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} a_n R^n. \quad (14.24)$$

(b) If $\operatorname{Ext} - \sum_{n=0}^{\infty} (-1)^n a_n R^n$ #-converges, then

$$\#- \lim_{x \to \# (x_0-R)+} f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} (-1)^n a_n R^n. \quad (14.25)$$

## 14.2. The $\mathbb{R}_c^\#$-valued #-exponential $\operatorname{Ext}\exp(x)$

We define the #-exponential $\operatorname{Ext}\exp(x)$ function as the solution of the differential equation

$$f^{\prime\#}(x) = f(x), f(0) = 1. \quad (14.26)$$

We solve it by setting

$$f(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} a_n x^n, f^{\prime\#}(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} n a_n x^n. \quad (14.27)$$

Therefore

$$\operatorname{Ext}\exp(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} \frac{x^n}{n!} \quad (14.28)$$

From Eq. (12.40) and Eq. (14.28) we get

$$(\operatorname{Ext}\exp(x))(\operatorname{Ext}\exp(y)) = \operatorname{Ext}\exp(x + y), \quad (14.29)$$

for any $x, y \in \mathbb{R}_c^\#$. We often denote #-exponential $\operatorname{Ext}\exp(x)$ by $\operatorname{Ext}\exp e^x$

$$\operatorname{Ext}\exp e^x. \quad (14.30)$$

## 14.3. The $\mathbb{R}_c^\#$-valued Trigonometric Functions $\operatorname{Ext}\sin(x)$ and $\operatorname{Ext}\cos(x)$.

We define the $\mathbb{R}_c^\#$-valued Trigonometric Functions $\operatorname{Ext}\sin(x)$ and $\operatorname{Ext}\cos(x)$ by

$$\operatorname{Ext}\sin(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \quad (14.31)$$

and

$$\operatorname{Ext}\cos(x) = \operatorname{Ext} - \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}. \quad (14.32)$$

It can be shown that the series (14.30) and (14.31) #-converge for all $x \in \mathbb{R}_c^\#$ and #-differentiating yields
$$ [\mathrm{Ext}\text{-}\sin(x)]'^{\#} = \mathrm{Ext}\text{-}\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = \mathrm{Ext}\text{-}\cos(x) \quad (14.33) $$

and

$$ \begin{aligned} [\mathrm{Ext}\text{-}\cos(x)]'^{\#} &= \mathrm{Ext}\text{-}\sum_{n=1}^{\infty} (-1)^n \frac{x^{2n-1}}{(2n-1)!} = -\mathrm{Ext}\text{-}\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = \\ &= -[\mathrm{Ext}\text{-}\sin(x)]. \end{aligned} \quad (14.34) $$

## 14.4. $\mathbb{R}_c^\#$-valued functions of several variables.

In this subsection we study $\mathbb{R}_c^\#$-valued functions defined on subsets of the $n$-dimensional external linear space $\mathbb{R}_c^{\#n}$, $n \in \mathbb{N}^\#$ which consists of all external and internal hyperfinite (or finite) sequences (called a vector) $\mathbf{X} = \{x_i\}_{i=1}^{i=n} = \{x_i\}_{i \in n}$ of hyperreal numbers, called the coordinates or components of vector $\mathbf{X}$.

**Definition 14.2.** The vector sum of $\mathbf{X} = \{x_i\}_{i=1}^{i=n}$ and $\mathbf{Y} = \{y_i\}_{i=1}^{i=n}$ is

$$ \mathbf{X} + \mathbf{Y} = \{x_i + y_i\}_{i=1}^{i=n}. \quad (14.35) $$

If $a \in \mathbb{R}_c^\#$ is a hyperreal number, the scalar multiple of $\mathbf{X}$ by $a$ is

$$ a \cdot \mathbf{X} = \{ax_i\}_{i=1}^{i=n}. \quad (14.36) $$

**Theorem 14.10.** If $\mathbf{X}, \mathbf{Y}$, and $\mathbf{Z}$ are in $\mathbb{R}_c^{\#n}$ and $a, b \in \mathbb{R}_c^\#$ are hyperreal numbers, then

(i) $\mathbf{X} + \mathbf{Y} = \mathbf{Y} + \mathbf{X}$ - vector addition is commutative

(ii) $(\mathbf{X} + \mathbf{Y}) + \mathbf{Z} = \mathbf{X} + (\mathbf{Y} + \mathbf{Z})$ - vector addition is associative

(iii) There is a unique vector $\mathbf{0}$, called the zero vector, such that $\mathbf{X} + \mathbf{0} = \mathbf{X}$ for all $\mathbf{X} \in \mathbb{R}_c^{\#n}$

(iv) For each $\mathbf{X} \in \mathbb{R}_c^{\#n}$ there is a unique vector $-\mathbf{X}$ such that $\mathbf{X} + (-\mathbf{X}) = \mathbf{0}$

(v) $a \cdot (b \cdot \mathbf{X}) = (ab) \cdot \mathbf{X}$

(vi) $(a+b) \cdot \mathbf{X} = a \cdot \mathbf{X} + b \cdot \mathbf{X}$

(vii) $a \cdot (\mathbf{X} + \mathbf{Y}) = a \cdot \mathbf{X} + a \cdot \mathbf{Y}$

(viii) $1 \cdot \mathbf{X} = \mathbf{X}$.

Clearly, $\mathbf{0} = \{0\}_{i=1}^{i=n}$ and, if $\mathbf{X} = \{x_i\}_{i=1}^{i=n}$, then $-\mathbf{X} = \{-x_i\}_{i=1}^{i=n}$.

We write $\mathbf{X} + (-\mathbf{Y})$ as $\mathbf{X} - \mathbf{Y}$. The point $\mathbf{0}$ is called the origin.

**Definition 14.3.** The length of the vector $\mathbf{X} = \{x_i\}_{i=1}^{i=n}$ is

$$ \|\mathbf{X}\| = (\mathrm{Ext}\text{-}\sum_{i=1}^{n} x_i^2)^{1/2}. \quad (14.37) $$

The distance between points $\mathbf{X}$ and $\mathbf{Y}$ is $\|\mathbf{X} - \mathbf{Y}\|$; in particular, $\|\mathbf{X}\|$ is the distance between $\mathbf{X}$ and the origin. If $\|\mathbf{X}\| = 1$, then $\mathbf{X}$ is a unit vector.

**Definition 14.4.** The inner product $\mathbf{X} \cdot \mathbf{Y}$ of $\mathbf{X} = \{x_i\}_{i=1}^{i=n}$ and $\mathbf{Y} = \{y_i\}_{i=1}^{i=n}$ is

$$ \mathbf{X} \cdot \mathbf{Y} = \mathrm{Ext}\text{-}\sum_{i=1}^{n} x_i y_i. \quad (14.38) $$

**Theorem 14.11.** (Schwarz's Inequality) If $\mathbf{X}, \mathbf{Y} \in \mathbb{R}_c^{\#n}$ then

$$ \|\mathbf{X} \cdot \mathbf{Y}\| \le \|\mathbf{X}\| \|\mathbf{Y}\|, \quad (14.39) $$

with equality if and only if one of the vectors is a scalar multiple of the other:

**Theorem 14.12.** (Triangle Inequality) If $\mathbf{X}, \mathbf{Y} \in \mathbb{R}_c^{\#n}$ then

$$ \|\mathbf{X} + \mathbf{Y}\| \le \|\mathbf{X}\| + \|\mathbf{Y}\|, \quad (14.40) $$
with equality if and only if one of the vectors is a nonnegative multiple of the other.

**Corollary 14.3.** If $X, Y, Z \in \mathbb{R}_c^{\#n}$, then

$$ \|\mathbf{X} - \mathbf{Z}\| \le \|\mathbf{X} - \mathbf{Y}\| + \|\mathbf{Y} - \mathbf{Z}\|. \quad (14.41) $$

**Corollary 14.4.** If $X, Y \in \mathbb{R}_c^{\#n}$, then

$$ \|\mathbf{X} - \mathbf{Y}\| \ge \|\mathbf{X}\| - \|\mathbf{Y}\|. \quad (14.42) $$

**Theorem 14.13.** If $X, Y, Z \in \mathbb{R}_c^{\#n}$ and $a \in \mathbb{R}_c^{\#}$ is a scalar, then

(i) $\|a\mathbf{X}\| = |a|\|\mathbf{X}\|$

(ii) $\|\mathbf{X}\| \ge 0$, with equality if and only if $\mathbf{X} = \mathbf{0}$

(iii) $\|\mathbf{X} - \mathbf{Y}\| \ge 0$, with equality if and only if $\mathbf{X} = \mathbf{Y}$

(iv) $\mathbf{X} \cdot \mathbf{Y} = \mathbf{Y} \cdot \mathbf{X}$

(v) $\mathbf{X} \cdot (\mathbf{Y} + \mathbf{Z}) = \mathbf{X} \cdot \mathbf{Y} + \mathbf{X} \cdot \mathbf{Z}$

(vi) $(c\mathbf{X}) \cdot \mathbf{Y} = \mathbf{X} \cdot (c\mathbf{Y}) = c(\mathbf{X} \cdot \mathbf{Y})$

**Definition 14.5.** Non-Archimedean metric space $(X, d)$ is a set $X$ together with a $\mathbb{R}_c^{\#}$-valued function $d: X \times X \to \mathbb{R}_c^{\#}$ (called a metric or non-Archimedean distance function) which assigns a hyperreal number $d(x, y)$ to every pair $x, y$ belongs $X$ satisfying the properties:

1. $d(x, y) \ge 0$ and $d(x, y) = 0$ iff $x = y$,

2. $d(x, y) = d(y, x)$,

3. $d(x, y) + d(y, z) \ge d(x, z)$.

**Remark 14.1.** Note that external linear space $\mathbb{R}_c^{\#n}$ endowed with distance function $d(\mathbf{X}, \mathbf{Y}) = \|\mathbf{X} - \mathbf{Y}\|$ satisfying the properties 1-3 mentioned above in Definition 14.5.

## 14.5. Line Segments in $\mathbb{R}_c^{\#n}, n \in \mathbb{N}^#.$

**Definition 14.6.** Suppose that $X_0, U \in \mathbb{R}_c^{\#n}$ and $U \ne \mathbf{0}$. Then the line through $X_0$ in the direction of $U$ is the set of all points in $\mathbb{R}_c^{\#n}$ of the form

$$ \mathbf{X}(\mathbf{X}_0, U) = \mathbf{X}_0 + tU, \quad -\infty < t < \infty^+. \quad (14.43) $$

A set of points of the form

$$ \mathbf{X} = \mathbf{X}_0 + t\mathbf{U}, \quad t_1 \le t \le t_2 \quad (14.44) $$

is called a line segment. In particular, the line segment from $X_0$ to $X_1$ is the set of points of the form

$$ \mathbf{X} = \mathbf{X}_0 + t(\mathbf{X}_1 - \mathbf{X}_0) = t\mathbf{X}_1 + (1-t)\mathbf{X}_0, \quad 0 \le t \le 1. \quad (14.45) $$

**Definition 14.7.** A hyper infinite sequence of points $X_n, n \in \mathbb{N}^# \text{ in } \mathbb{R}_c^{\#n}$ #-converges to the #-limit $\bar{\mathbf{X}}$ if

$$ \#\text{-lim}_{n \to \# \infty^#} \|\mathbf{X}_n - \bar{\mathbf{X}}\| = 0. \quad (14.46) $$

In this case we write $\#\text{-lim}_{n \to \# \infty^#} \mathbf{X}_n = \bar{\mathbf{X}}$.

**Theorem 14.14.** Let $\bar{\mathbf{X}} = \{x_i\}_{i=1}^{i=n}$ and $\mathbf{X}_m = \{x_{i_m}\}_{i=1}^{i=n}, m \ge 1$. Then $\#\text{-lim}_{m \to \# \infty^#} \mathbf{X}_m = \bar{\mathbf{X}}$ if and only if $\#\text{-lim}_{m \to \# \infty^#} x_{i_m} = \bar{x}_i, 1 \le i \le n$; that is a hyper infinite sequence $\{\mathbf{X}_m\}$ of points in $\mathbb{R}_c^{\#n}$ #-converges to a #-limit $\bar{\mathbf{X}}$ if and only if the hyper infinite sequences of components of $\{\mathbf{X}_m\}$ #-converge to the respective components of $\bar{\mathbf{X}}$.

**Theorem 14.15.** (Cauchy's #-Convergence Criterion) A hyper infinite sequence $\{\mathbf{X}_m\}$ in $\mathbb{R}_c^{\#n}$ #-converges if and only if for each $\varepsilon > 0, \varepsilon \approx 0$, there is an hyperinteger $N \in \mathbb{N}^n \setminus \mathbb{N}$ such that
$$ \| \mathbf{X}_n - \mathbf{X}_m \| < \varepsilon \qquad (14.47) $$

if $n, m \ge N$.

**Definition 14.8.** If A is a subset of a metric space $\mathbb{R}_c^{\#n}$ then $x$ is a \#-limit point of A if it is the \#-limit of an eventually non-constant hyper infinite sequence $\{a_i\}_{i \in \mathbb{N}^\#}$ of points of A.

**Definition 14.9.** A subset A is said to be a \#-closed subset of $\mathbb{R}_c^{\#n}$ if it contains all its \#-limit points.

**Example 14.1.** (i) $\mathbb{R}_c^\#$ with the canonical metric $d(x, y) = |x - y|$, since in $\mathbb{R}_c^\#$ every hyperreal number is a \#-limit point of the hyper infinite sequence $\{q_i\}_{i \in \mathbb{N}^\#}$ of hyperrationals $q_i \in \mathbb{Q}^\#$, $i \in \mathbb{N}^\#$.

(ii) The empty set is \#-closed.

(iii) Any finite set is \#-closed.

(iv) Any hyperfinite set is \#-closed.

(v) The closed interval $[a, b]$, where $a, b \in \mathbb{R}_c^\#$, is \#-closed subset of $\mathbb{R}_c^\#$ with its canonical metric.

(vi) Let $\Delta$ be a set $\Delta = \{\varepsilon \mid \varepsilon \mid \approx 0\}$. A set $\Delta$ is \#-closed subset of $\mathbb{R}_c^\#$, since in $\Delta$ every hyperreal number $\delta \in \Delta$ is a \#-limit point of the hyper infinite sequence $\{q_i\}_{i \in \mathbb{N}^\#}$ of hyperrationals $q_i \in \Delta \cap \mathbb{Q}^\#$, $i \in \mathbb{N}^\#$.

**Definition 14.10.** An \#-neighbourhood of a point $p$ in a metric space $(X, d)$ is the set $N_\varepsilon(p) = \{x \in X \mid d(x, p) < \varepsilon, \varepsilon \approx 0\}$

**Definition 14.11.** A subset A of a metric space $(X, d)$ is called \#-open in X if every point of A has an \#-neighbourhood which lies completely in A.

**Example 14.2.** (i) Any open interval $(a, b)$ is an \#-open set in $\mathbb{R}_c^\#$ with its canonical metric $d(x, y) = |x - y|$.

(ii) A set $\Delta = \{\varepsilon \mid \varepsilon \mid \approx 0\}$ is \#-open subset of $\mathbb{R}_c^\#$, since every point of $\Delta$ obviously has a \#-neighbourhood which lies completely in $\Delta$.

**Remark 14.2.** Note that a set $\Delta = \{\varepsilon \mid \varepsilon \mid \approx 0\}$ are \#-open and \#-closed simultaneously.

**Definition 14.12.** A subset A of a non-Archimedean metric space X is *admissible* if A is exactly \#-closed or exactly \#-open but not \#-open and \#-closed simultaneously.

**Theorem 14.16.** (i) The union (of an arbitrary number) of \#-open admissible sets is \#-open. (ii) The intersection of finitely or hyper finitely many \#-open admissible sets is \#-open.

**Proof.** (i) Let $x \in \bigcup A_i = A$. Then $x \in A_i$ for some $i$. Since this is \#-open, $x$ has an \#-neighbourhood lying completely inside $A_i$ and this is also inside A.

(ii) It is enough to show this for just two \#-open sets A and B. So suppose $x \in A \cap B$. Then $x \in A$ and so has an \#-neighbourhood $N_{\varepsilon_1}(p), \varepsilon_1 \approx 0$ lying in A. Similarly $x$ has an \#-neighbourhood $N_{\varepsilon_2}(p), \varepsilon_2 \approx 0$ lying in B. So if $\varepsilon = \min\{\varepsilon_1, \varepsilon_2\}$ the \#-neighbourhood $N_\varepsilon(p)$ lies in both A and B and hence in $A \cap B$. By hyper infinite induction statement (ii) holds in general.

**Theorem 14.17.** Any admissible subset A of a metric space X is \#-closed if and only if its complement $X\setminus A$ is admissible and is \#-open subset of a metric space X.

**Proof.** 1. Suppose A is admissible and A is \#-closed. We need to show that $X\setminus A$ is \#-open.

So suppose that $x$ belongs $X\setminus A$. Then some \#-neighbourhood of $x$ does not meet A (otherwise $x$ would be a \#-limit point of A and hence in A). Thus this \#-neighbourhood
of $x$ lies completely in $X\setminus A$ which is what we needed to prove.

2. Conversely, suppose that $X\setminus A$ is #-open. We need to show that $A$ contains all its #-limit points. So suppose $x$ is a #-limit point of $A$ and that $x \notin A$. Then $x \in X\setminus A$ and hence

has an #-neighbourhood subset $X\setminus A$. But this is an #-neighbourhood that does not meet $A$

and we have a contradiction.

**Definition 14.13.** If $S$ is a nonempty subset of $\mathbb{R}_c^{\#n}$, then

$$d(S) = \sup\{\|\mathbf{X} - \mathbf{Y}\|\mid\mathbf{X}, \mathbf{Y} \in S\} \qquad (14.48)$$

is the diameter of $S$. If $d(S) < \infty^{\#}$, $S$ is bounded or hyperbounded. If $d(S) = \infty^{\#}$, $S$ is hyperunbounded.

**Theorem 14.18.** (Principle of Nested Sets) If $S_1, S_2, \dots$, are #-closed nonempty subsets of $\mathbb{R}_c^{\#n}$ such that

$$\forall r(r \in \mathbb{N}^{\#})[S_{r+1} \subset S_r] \qquad (14.49)$$

and

$$\#\lim_{r \to \infty} d(S_r) = 0, \qquad (14.50)$$

then the intersection

$$\Lambda = \bigcap_{r=1}^{\infty} S_r \qquad (14.51)$$

contains exactly one point:

**Proof.** Let $\{\mathbf{X}_r\}$ be a hyper infinite sequence such that $\mathbf{X}_r \in S_r, r \ge 1$. Because of (14.49), $\mathbf{X}_r \in S_k$ if $r \ge k$, so $\|\mathbf{X}_r - \mathbf{X}_s\| < d(S_k)$ if $r, s \ge k$.

From (14.50) and Theorem 14.15., $\mathbf{X}_r$ #-converges to a #-limit $\bar{\mathbf{X}}$. Since $\bar{\mathbf{X}}$ is a #-limit point of every $S_k$ and every $S_k$ is #-closed, $\bar{\mathbf{X}}$ is in every $S_k$. Therefore, $\bar{\mathbf{X}} \in \Lambda$, so $\Lambda \ne \emptyset$.

Moreover, $\bar{\mathbf{X}}$ is the only point in $\Lambda$, since if $\mathbf{Y} \in \Lambda$, then $\|\bar{\mathbf{X}} - \mathbf{Y}\| < d(S_k), k \ge 1$, and (14.50) implies that $\mathbf{Y} = \bar{\mathbf{X}}$.

**Definition 14.14.** If $S$ is a nonempty admissible subset of $\mathbb{R}_c^{\#n}$ we say that $S$ is a #-compact set in $\mathbb{R}_c^{\#n}$ if is a #-closed and bounded or hyperbounded set.

**Definition 14.15.** Collection $\mathbf{H}$ of admissible #-open sets is an #-open covering of a set $S$ if $S \subset \cup\{\mathbf{H}|\mathbf{H} \in \mathbf{H}\}$.

**Theorem 14.19.** (Heine-Borel Theorem) If $\mathbf{H}$ is an #-open covering of a #-compact subset $S$, then $S$ can be covered by hyper finitely many sets from $\mathbf{H}$.

**Proof.** The proof is by contradiction. We first consider the case where $n = 2$. Suppose that there is a #-open covering $\mathbf{H}$ for $S$ from which it is impossible to select a hyperfinite

subcovering. Since $S$ is bounded or hyperbounded, $S$ is contained in a #-closed square $T = \{\{x, y\}|a_1 \le x \le a_1 + L, a_2 \le ya_2 + L\}$ with sides of length $L$ (Pic. 14.5.1).
<figure><img src="image_1.png" /><figcaption>Pic.14.5.1.</figcaption></figure>

Bisecting the sides of T as shown by the dashed lines in Figure 14.5.1 leads to four #-closed squares, $T^{(1)}, T^{(2)}, T^{(3)},$ and $T^{(4)}$, with sides of length $L/2$. Let $S^{(i)} = S \cap T$, $1 \le i \le 4$. Each $S^{(i)}$, being the intersection of admissible #-closed sets, is #-closed, and $S = \cup_{i=1}^4 S^{(i)}$. Moreover, $H$ covers each $S^{(i)}$, but at least one $S^{(i)}$ cannot be covered by any finite or hyperfinite subcollection of $H$, since if all the $S^{(i)}$ could be, then so could $S$. Let $S_1$ be a set with this property, chosen from $S^{(1)}, S^{(2)}, S^{(3)},$ and $S^{(4)}$.

We are now back to the situation we started from: a #-compact set $S_1$ covered by $H$, but not by any hyperfinite subcollection of $H$. However, $S_1$ is contained in a square $T_1$ with sides of length $L/2$ instead of $L$. Bisecting the sides of $T_1$ and repeating the argument, we obtain a subset $S_2$ of $S_1$ that has the same properties as $S$, except that it is contained in a square with sides of length $L/4$. Continuing in this way produces a hyper infinite sequence of nonempty #-closed sets $S_0 = S, S_1, S_2, \dots$, such that $S_k \supset S_{k+1}$ and $d(S_k) \le L/2^{k-1/2}, k \ge 0$. From Theorem 14.18, there is a point $\bar{X}$ in $\cap_{k=1}^\infty S_k$. Since $\bar{X} \in S$, there is an open set $H$ in $H$ that contains $\bar{X}$, and this $H$ must also contain some #-neighborhood of $\bar{X}$. Since every $X$ in $S_k$ satisfies the inequality $||X - \bar{X}|| < 2^{-k+1/2}L$, it follows that $S_k \subset H$ for $k \in \mathbb{N}^\# / \mathbb{N}$ sufficiently large. This contradicts our assumption on $H$, which led us to believe that no $S_k$ could be covered by a hyperfinite

number of sets from $H$. Consequently, this assumption must be false: $H$ must have a finite or hyperfinite subcollection that covers $S$. This completes the proof for $n = 2$. The idea of the proof is the same for $n > 2$. The counterpart of the square $T$ is the hypercube with sides of length $L$:

$$T = \{(x_1, x_2, \dots, x_n) | a_i \le x_i \le a_i + L, 1 \le i \le n\}.$$

Halving the intervals of variation of the $n$ coordinates $x_1, x_2, \dots, x_n$ divides $T$ into $2^n$ closed hypercubes with sides of length $L/2$:

$$T^{(i)} = \{(x_1, x_2, \dots, x_n) | b_i \le x_i \le b_i + L/2, 1 \le i \le n\},$$

where $b_i = a_i$ or $b_i = a_i + L/2$. If no hyperfinite subcollection of $H$ covers $S$, then at least one of these smaller hypercubes must contain a subset of $S$ that is not covered by any hyperfinite subcollection of $S$. Now the proof proceeds as for $n = 2$.

## 14.6. #-Neighborhoods and #-open sets in $\mathbb{R}_c^{\#n}$, $n \in \mathbb{N}^\#$.
Connected Sets and Regions in $\mathbb{R}_c^{\#n}$.

**Definition 14.16.** Assum that $A$ is admissible subset of $\mathbb{R}_c^{\#n}$.

(i) The #-interior $\text{-int}(A)$ of a set $A$ is the largest open subset $A$,
(ii) The #-closure #-cl(A) of a set A is the smallest #-closed set containing A.

**Theorem 14.20.** 1. #-cl(Ø) = Ø

2. A ⊂ #-cl(A) for any set A.

3. #-cl(A ∪ B) = #-cl(A) ∪ #-cl(B) for any sets A and B.

4. cl(#-cl(A)) = #-cl(A) for any set A.

**Proof.**1. and 2. follow from the definition.

To prove 3 note that #-cl(A) ∪ #-cl(B) is a #-closed set which contains A ∪ B and so
#-cl(A) ⊂ #-cl(A ∪ B). Similarly, #-cl(B) ⊂ #-cl(A ∪ B) and so
#-cl(A) ∪ #-cl(B) ⊂ #-cl(A ∪ B) and the result follows.

To prove 4 we have #-cl(A) ⊂ #-cl(#-cl(A)) from 2. Also #-cl(A) is a #-closed set which contains #-cl(A) and hence it contains #-cl(#-cl(A)).

**Example 14.3.**For ℝ<sup>#n</sup> with its usual topology induced by its canonical metric
d(x,y) = |x - y|, #-cl((a,b)) = [a,b] and #-int([a,b]) = (a,b).

**Definition 14.17.** If $\varepsilon > 0, \varepsilon \approx 0$, the $\varepsilon$-neighborhood of a point $X_0$ in $\mathbb{R}_c^{\#n}$ is the set

$$
N_{\varepsilon}(\mathbf{X}_0) = \{\mathbf{X} \mid \|\mathbf{X} - \mathbf{X}_0\| < \varepsilon\}. \qquad (14.52)
$$

**Definition 14.18.**If $X_0$ is a point in in $\mathbb{R}_c^{\#n}$ and $r > 0$, the sphere of radius $r$ about $X_0$ is the set $S_r(X_0) = \{\mathbf{X} \mid \|\mathbf{X} - X_0\| = r\}$

**Definition 14.19.**If $X_0$ is a point in in $\mathbb{R}_c^{\#n}$ and $r > 0$, the #-open $n$-ball of radius $r$ about $X_0$ is the set $B_r(X_0) = \{\mathbf{X} \mid \|\mathbf{X} - X_0\| < r\}$. Thus, $\varepsilon$-neighborhoods are #-open $n$-balls.
If $X_1$ is in $B_r(X_0)$ and $\|\mathbf{X} - X_1\| < \varepsilon = r - \|\mathbf{X} - X_0\|$, then $X$ is in $B_r(X_0)$.

Thus, $B_r(X_0)$ contains an $\epsilon$-neighborhood of each of its points, and is therefore
\#-open.

The #-closure of $B_r(X_0)$ is the #-closed $n$-ball of radius $r$ about $X_0$, defined by
\#-cl($B_r(X_0)$) = $\{\mathbf{X} \mid \|\mathbf{X} - X_0\| < r\}$, $r = \|X_1 - X_0\|$.

**Proposition14.1.** If $X_1$ and $X_2$ are in $S_r(X_0)$ for some $r > 0$, then so is every point on
the line segment from $X_1$ to $X_2$.

**Definition 14.20.** A subset $S \subset \mathbb{R}_c^{\#n}$ is #-connected if it is impossible to represent $S$ as the union of two disjoint nonempty sets such that neither contains a #-limit point of the other; that is, if $S$ cannot be expressed as $S = A \cup B$, where

$$
A \neq \emptyset, B \neq \emptyset, \#-cl(A) \cap B = \emptyset, \#-cl(B) \cap A = \emptyset. \quad (14.53)
$$

If $S$ can be expressed in this way, then $S$ is #-disconnected.

**Definition 14.21.** A region $S$ in $\mathbb{R}_c^{\#n}$ is the union of an #-open #-connected set with some, all, or none of its #-boundary; thus, #-int($S$) is #-connected, and every point of $S$ is a #-limit point of #-int($S$).

14.7. The #-limits and #-continuity $\mathbb{R}_c^#$-valued functions of
$n \in \mathbb{N}^#$ variables.

We denote the domain of a function $f$ by $D_f$ and the value of $f$ at a point $X = \{x_i\}_{i=1}^{i=n}$ by $f(X)$ or $f(\{x_i\}_{i=1}^{i=n})$.

**Definition 14.22.** We say that $f(\mathbf{X})$ #-approaches the #-limit $L$ as $\mathbf{X}$ #-approaches $\mathbf{X}_0$
and write

$$
\#\text{-lim}_{\mathbf{X} \to \#} \mathbf{x}_0 f(\mathbf{X}) = L \qquad (14.54)
$$

if $X_0$ is a #-limit point of $D_f$ and, for every $\varepsilon > 0$, $\varepsilon \approx 0$, there is a $\delta > 0$, $\delta \approx 0$, such that $|f(\mathbf{X}) - L| < \varepsilon$ for all $\mathbf{X} \in D_f$ such that $0 < \|\mathbf{X} - \mathbf{X}_0\| < \delta$.
**Theorem 14.21.** If $\#\lim_{X \to \#} x_0 f(X)$ exists, then it is unique.

**Theorem 14.22.** Suppose that $f$ and $g$ are defined on a set $D \subset \mathbb{R}_c^{\#n}$, $X_0$ is a $\#$-limit point of $D$, and $\#\lim_{X \to \#} x_0 f(X) = L_1$, $\#\lim_{X \to \#} x_0 g(X) = L_2$. Then

$$ \begin{aligned} \#\lim_{X \to \#} x_0 (f \pm g)(X) &= L_1 \pm L_2, \\ \#\lim_{X \to \#} x_0 (f \times g)(X) &= L_1 L_2, \end{aligned} \tag{14.55} $$ 
and 
$$ \#\lim_{X \to \#} x_0 (f/g)(X) = L_1 L_2, $$

if $L_2 \neq 0$.

**Definition 14.23.** We say that $f(X)$ $\#$-approaches $\infty^\#$ as $X$ $\#$-approaches $X_0$ and write

$$ \#\lim_{X \to \#} x_0 f(X) = \infty^\# \tag{14.56} $$

if $X_0$ is a $\#$-limit point of $D_f$ and, for every hyperreal number $M$, there is a $\delta > 0$, $\delta \approx 0$, such that $f(X) > M$ whenever $0 < \|X - X_0\| < \delta$ and $X \in D_f$. We say that

$$ \#\lim_{X \to \#} x_0 f(X) = -\infty^\# \tag{14.57} $$

if $\#\lim_{X \to \#} x_0 [-f(X)] = \infty^\#$.

**Definition 14.24.** If $D_f$ is hyperunbounded, we say that

$$ \#\lim_{\|X\| \to \#} \infty^\# f(X) = L, \tag{14.58} $$

where $L$ finite or hyperfinite if for every $\delta > 0$, $\delta \approx 0$, there is a number $R \in \mathbb{R}_c^\#$ such that $|f(X) - L| < \varepsilon$ whenever $\|X\| > R$ and $X \in D_f$.

**Definition 14.25.** If $X_0 \in D_f$ and is a $\#$-limit point of $D_f$, then we say that $f$ is $\#$-continuous at $X_0$ if

$$ \#\lim_{X \to \#} x_0 f(X) = f(X_0). \tag{14.59} $$

**Theorem 14.23.** Suppose that $X_0 \in D_f$ and is a $\#$-limit point of $D_f$. Then $f$ is $\#$-continuous at $X_0$ if and only if for each $\varepsilon > 0$, $\varepsilon \approx 0$ there is a $\delta > 0$, $\delta \approx 0$ such that $|f(X) - f(X_0)| < \varepsilon$ whenever $\|X - X_0\| < \delta$ and $X \in D_f$.

**Definition 14.26.** We will say that $f$ is $\#$-continuous on $S$ if $f$ is $\#$-continuous at every point of $S$.

**Theorem 14.24.** If $f$ and $g$ are $\#$-continuous on a set $S \subset \mathbb{R}_c^{\#n}$, then so are $f \pm g$, and $fg$. Also, $f/g$ is $\#$-continuous at each $X_0 \in S$ such that $g(X_0) \neq 0$.

**Definition 14.27.** Suppose that $g_1, g_2, \dots, g_n, n \in \mathbb{N}^\#$ are $\mathbb{R}_c^\#$-valued functions defined on a subset $T \subset \mathbb{R}_c^{\#n}$, and define the vector-valued function $G$ on $T$ by

$$ G(U) = (g_1(U), g_2(U), \dots, g_n(U)), U \in T. \tag{14.60} $$

Then $g_1, g_2, \dots, g_n$ are the component functions of $G = (g_1, g_2, \dots, g_n)$. We say that

$$ \#\lim_{U \to \#} U_0 G(U) = L = (L_1, \dots, L_n) \tag{14.61} $$

if $\#\lim_{U \to \#} U_0 g_i(U) = L_i$, $1 \le i \le n$ and that $G$ is $\#$-continuous at $U_0$ if $g_1, g_2, \dots, g_n$ are each $\#$-continuous at $U_0$.

**Theorem 14.25.** For a vector-valued function $G$, $\#\lim_{U \to \#} U_0 G(U) = L$ if and only if for each $\varepsilon > 0$, $\varepsilon \approx 0$ there is a $\delta > 0$, $\delta \approx 0$ such that $\|G(U) - L\| < \varepsilon$ whenever $0 < \|U - U_0\| < \delta$ and $U \in D_G$. Similarly, $G$ is $\#$-continuous at $U_0$ if and only if for
each $\varepsilon > 0, \varepsilon \approx 0$ there is a $\delta > 0, \delta \approx 0$ such that $\|G(\mathbf{U}) - G(\mathbf{U}_0)\| < \varepsilon$ whenever $\|\mathbf{U} - \mathbf{U}_0\| < \delta$ and $\mathbf{U} \in \mathbf{D}_G$.

**Theorem 14.26.** Let $f$ be a $\mathbb{R}_c^\#$-valued function defined on a subset of $\mathbb{R}_c^{\#n}$, and let the vector-valued function $G = (g_1, g_2, \dots, g_n)$ be defined on a domain $\mathbf{D}_G$ in $\mathbb{R}_c^{\#n}$. Let the set $T = \{\mathbf{U}|\mathbf{U} \in \mathbf{D}_G \text{ and } G(\mathbf{U}) \in D_f\}$, be nonempty; and define the $\mathbb{R}_c^\#$-valued composite function $h = f \circ G$ on $T$ by $h(\mathbf{U}) = f(G(\mathbf{U}))$, $\mathbf{U} \in T$. Now suppose that $\mathbf{U}_0 \in T$ and is a #-limit point of $T$, $G$ is #-continuous at $\mathbf{U}_0$, and $f$ is #-continuous at $\mathbf{X}_0 = G(\mathbf{U}_0)$. Then $h$ is #-continuous at $\mathbf{U}_0$.

**Theorem 14.27.** If $f$ is #-continuous on a #-compact set $S \subset \mathbb{R}_c^{\#n}$, then $f$ is bounded or hyperbounded on $S$.

**Theorem 14.28.** Let $f$ be #-continuous on a compact set $S \subset \mathbb{R}_c^{\#n}$ and $\alpha = \inf_{\mathbf{X} \in S} f(\mathbf{X})$, $\beta = \sup_{\mathbf{X} \in S} f(\mathbf{X})$. Then $f(\mathbf{X}_1) = \alpha$ and $f(\mathbf{X}_2) = \beta$ for some $\mathbf{X}_1$ and $\mathbf{X}_2$ in $S$.

**Theorem 14.29.** (Intermediate Value Theorem) Let $f$ be #-continuous on a region $S \subset \mathbb{R}_c^{\#n}$. Suppose that $\mathbf{A}$ and $\mathbf{B}$ are in $S$ and $f(\mathbf{A}) < u < f(\mathbf{B})$. Then $f(\mathbf{C}) = u$ for some $\mathbf{C} \in S$.

**Definition 14.28.** $f$ is uniformly #-continuous on a subset $S$ of its domain in $\mathbb{R}_c^{\#n}$ if for every $\varepsilon > 0, \varepsilon \approx 0$ there is a $\delta > 0, \delta \approx 0$ such that $|f(\mathbf{X}) - f(\mathbf{X}_0)| < \varepsilon$ whenever $\|\mathbf{X} - \mathbf{X}_0\| < \delta$ and $\mathbf{X}, \mathbf{X}_0 \in S$.

**Theorem 14.30.** If $f$ is #-continuous on a #-compact set $S \subset \mathbb{R}_c^{\#n}$ then $f$ is uniformly #-continuous on $S$.

## 14.8. Partial #-Derivatives and the #-Differential

**Definition 14.29.** Let $\Phi$ be a unit vector and $\mathbf{X}$ a point in $\mathbb{R}_c^{\#n}$. The directional #-derivative of $f(\mathbf{X})$ at $\mathbf{X}$ in the direction of $\Phi$ is defined by

$$ \frac{\partial'^{\#} f(\mathbf{X})}{\partial'^{\#} \Phi} = \#- \lim_{t \to \# 0} \frac{f(\mathbf{X} + t\Phi) - f(\mathbf{X})}{t} \quad (14.62) $$

if the #-limit exists. That is, $\partial'^{\#} f(\mathbf{X})/\partial'^{\#} \Phi$ is the ordinary derivative of the function $H(t) = f(\mathbf{X} + t\Phi)$ at $t = 0$, if $H'^{(t)}$ exists. The directional #-derivatives that we are most interested in are those in the directions of the unit vectors $\mathbf{E}_i$, $1 \le i \le n$, where all components of $\mathbf{E}_i$ are zero except for the $i$-th, which is 1.

**Definition 14.30.** Since $\mathbf{X}$ and $\mathbf{X} + t\mathbf{E}_i$ differ only in the $i$-th coordinate, $\partial'^{\#} f(\mathbf{X})/\partial'^{\#} \mathbf{E}_i$ is called the partial #-derivative of $f$ with respect to $\mathbf{x}_i$ at $\mathbf{X}$. It is also denoted by $\partial'^{\#} f(\mathbf{X})/\partial'^{\#} \mathbf{x}_i$ or $f_{\mathbf{x}_i}^{\#}(\mathbf{X})$, thus,

$$ \frac{\partial'^{\#} f(\mathbf{X})}{\partial'^{\#} \mathbf{x}_i} = f_{\mathbf{x}_i}^{\#}(\mathbf{X}) = \#- \lim_{t \to \# 0} \frac{f(\{\mathbf{x}_i + t\}_{i \in n}) - f(\{\mathbf{x}_i\}_{i \in n})}{t} \quad (14.63) $$

If $\mathbf{X} = (x, y)$, then we denote the partial #-derivatives accordingly; thus,

$$ \frac{\partial'^{\#} f(x, y)}{\partial'^{\#} x} = f_x^{\#}(x, y) = \#- \lim_{h \to \# 0} \frac{f(x + h, y) - f(x, y)}{h} \quad (14.64) $$

and

$$ \frac{\partial'^{\#} f(x, y)}{\partial'^{\#} y} = f_y^{\#}(x, y) = \#- \lim_{h \to \# 0} \frac{f(x, y + h) - f(x, y)}{h} \quad (14.65) $$

**Theorem 14.31.** If $f_{\mathbf{x}_i}^{\#}(\mathbf{X})$ and $g_{\mathbf{x}_i}^{\#}(\mathbf{X})$ exist, then
$$ \frac{\partial'^{\#}(f+g)(\mathbf{X})}{\partial'^{\#}x_i} = f_{x_i}^{\#}(\mathbf{X}) + g_{x_i}^{\#}(\mathbf{X}), \quad \frac{\partial'^{\#}(f \times g)(\mathbf{X})}{\partial'^{\#}x_i} = f_{x_i}^{\#}(\mathbf{X})g(\mathbf{X}) + g_{x_i}^{\#}(\mathbf{X})f(\mathbf{X}), \quad (14.66) $$

and, if $g(\mathbf{X}) \neq 0$,

$$ \frac{\partial'^{\#}(f/g)(\mathbf{X})}{\partial'^{\#}x_i} = \frac{g(\mathbf{X})f_{x_i}^{\#}(\mathbf{X}) - f(\mathbf{X})g_{x_i}^{\#}(\mathbf{X})}{[g(\mathbf{X})]^2}. \quad (14.67) $$

If $f_{x_i}^{\#}(\mathbf{X})$ exists at every point of a set $D \subset \mathbb{R}_c^{\#n}$, then it defines a function $f_{x_i}^{\#}(\mathbf{X})$ on $D$. If this function has a partial #-derivative with respect to $x_j$ on a subset of $D$, we denote the partial #-derivative by

$$ \frac{\partial'^{\#}f(\mathbf{X})}{\partial'^{\#}x_j} \left( \frac{\partial'^{\#}f(\mathbf{X})}{\partial'^{\#}x_i} \right) = \frac{\partial'^{2\#}f(\mathbf{X})}{\partial'^{\#}x_j \partial'^{\#}x_i} = f_{x_i x_j}^{\#}(\mathbf{X}). \quad (14.68) $$

The function obtained by differentiating $f(\mathbf{X})$ successively with respect to $x_{i_1}, x_{i_2}, \dots, x_{i_r}$ is denoted by

$$ \frac{\partial^{r'\#}f(\mathbf{X})}{\partial'^{\#}x_{i_1} \partial'^{\#}x_{i_{r-1}} \dots \partial'^{\#}x_{i_r}} = f_{x_{i_1} \dots x_{i_{r-1}} x_{i_r}}^{\#}(\mathbf{X}) \quad (14.69) $$

it is an $r$-th-order partial derivative of $f(\mathbf{X})$.

**Theorem 14.32.** Suppose that $f, f_x^{\#}, f_y^{\#}$, and $f_{xy}^{\#}$ exist on a #-neighborhood $\Omega$ of $(x_0, y_0)$, and $f_{xy}^{\#}$ is #-continuous at $(x_0, y_0)$. Then $f_{yx}^{\#}$ $(x_0, y_0)$ exists, and

$$ f_{yx}^{\#}(x_0, y_0) = f_{xy}^{\#}(x_0, y_0). \quad (14.70) $$

**Theorem 14.33.** Suppose that $f$ and all its partial #-derivatives of order $r$ are #-continuous on an #-open subset $S$ of $\mathbb{R}_c^{\#n}$. Then

$$ f_{x_{i_1}, x_{i_2}, \dots, x_{i_r}}^{\#}(\mathbf{X}) = f_{x_{j_1}, x_{j_2}, \dots, x_{j_r}}^{\#}(\mathbf{X}), \mathbf{X} \in S, \quad (14.71) $$

if each of the variables $x_1, x_2, \dots, x_n$ appears the same number of times in $\{x_{i_1}, x_{i_2}, \dots, x_{i_r}\}$ and $\{x_{j_1}, x_{j_2}, \dots, x_{j_r}\}$. If this number is $r_k$, we denote the common value of the two sides of (14.71) by

$$ \frac{\partial^{r'\#}f(\mathbf{X})}{\partial'^{\#}x_{i_r} \partial'^{\#}x_{i_{r-1}} \dots \partial'^{\#}x_{i_1}}. \quad (14.72) $$

**Definition 14.31.** A function $f(\mathbf{X})$ is #-differentiable at $\mathbf{X}_0 = (x_{10}, x_{20}, \dots, x_{n0})$ if there are constants $m_1, m_2, \dots, m_n$ such that

$$ \#\text{-lim}_{\|\mathbf{x}-\mathbf{x}_0\| \to 0} \frac{f(\mathbf{x}) - f(\mathbf{x}_0) - \left( \text{Ext} - \sum_{i=1}^n m_i (\mathbf{x}_i - \mathbf{x}_0) \right)}{\|\mathbf{x}-\mathbf{x}_0\|} = 0. \quad (14.73) $$

**Theorem 14.34.** If $f$ is differentiable at $\mathbf{X}_0 = (x_{10}, x_{20}, \dots, x_{n0})$, then $f_{x_0}^{\#}(\mathbf{X}_0)$, $1 \le i \le n$, exist and the constants $m_i$, $1 \le i \le n$, in Eq. (14.73) are given by

$$ m_i = f_{x_{i0}}^{\#}(\mathbf{X}_0). \quad (14.74) $$

**Theorem 14.35.** If $f$ is #-differentiable at $\mathbf{X}_0$, then $f$ is #-continuous at $\mathbf{X}_0$.

**Definition 14.32.** A linear function $L: \mathbb{R}_c^{\#n} \to \mathbb{R}^\#$ is a $\mathbb{R}_c^\#$-valued function of the form

$$ L(\mathbf{X}) = \text{Ext} - \sum_{i=1}^{n} m_i x_i, \quad (14.75) $$

where $m_i$, $1 \le i \le n$ are constants. From Definition 14.31, $f$ is #-differentiable at $\mathbf{X}_0$ if and only if there is a linear function $L$ such that $f(\mathbf{X}) - f(\mathbf{X}_0)$ can be approximated so well near $\mathbf{X}_0$ by

$$ f(\mathbf{X}) - f(\mathbf{X}_0) = L(\mathbf{X} - \mathbf{X}_0) + E(\mathbf{X}) \|\mathbf{X} - \mathbf{X}_0\|, \quad (14.76) $$
where

$$ \#\lim_{\|\mathbf{x}-\mathbf{x}_0\| \to 0} E(\mathbf{x}) = 0. \qquad (14.77) $$

**Remark 14.3.** Theorem 14.34 implies that if $f$ is \#-differentiable at $\mathbf{X}_0$, then there is exactly one linear function $L$ that satisfies (14.76) and (14.77).

This function is called the \#-differential of $f$ at $\mathbf{X}_0$. We will denote it by $d_{\mathbf{X}_0}^{\#}f$ and its value by $(d_{\mathbf{X}_0}^{\#}f)(\mathbf{X})$; thus,

$$ (d_{\mathbf{X}_0}^{\#}f)(\mathbf{X}) = \mathrm{Ext} - \sum_{i=1}^{n} f_{x_{i0}}^{\#}(\mathbf{X}_0)x_i. \qquad (14.78) $$

For convenience in writing $d_{\mathbf{X}_0}^{\#}f$, and to conform with standard notation, we introduce the function $d^{\prime\#}x_i : \mathbb{R}_c^{\#n} \to \mathbb{R}_c^{\#} \text{ defined by } dx_i(\mathbf{X}) = x_i$. That is, $d^{\prime\#}x_i$ is the function whose value at a point in $\mathbb{R}_c^{\#n}$ is the $i$-th coordinate of the point. It is the \#-differential of the function $g_i(\mathbf{X}) = x_i$. From Eq. (14.78)

$$ d_{\mathbf{X}_0}^{\#}f = \mathrm{Ext} - \sum_{i=1}^{n} f_{x_{i0}}^{\#}(\mathbf{X}_0)d^{\prime\#}x_i. \qquad (14.78) $$

## 15.\#-Analytic functions $f : \mathbb{C}_c^\# \to \mathbb{C}_c^\#.$

### 15.1. $\mathbb{C}_c^\#$-valued \#-analytic functions $f : \mathbb{C}_c^\# \to \mathbb{C}_c^\#$.

The class of \#-analytic functions is formed by the complex functions of a complex variable $z \in \mathbb{C}_c^\# = \mathbb{R}_c^\# + i\mathbb{R}_c^\#$ which possess a \#-derivative wherever the function is defined. The term \#-holomorphic function is used with identical meaning. For the purpose of this preliminary investigation the reader may think primarily of functions which are defined in the whole plane $\mathbb{C}_c^\#$.

The definition of the \#-derivative can be written in the form

$$ f^{\prime\#}(z) = \#\lim_{h \to \#} \lim_{h \to 0} \frac{f(z+h) - f(z)}{h} \qquad (15.1) $$

As a first obvious consequence $f(z)$ is necessarily \#-continuous. Indeed, from $f(z+h) - f(z) = h \times (f(z+h) - f(z))/h$ one obtains $\#\lim_{h \to \#} \lim_{h \to 0} (f(z+h) - f(z)) = 0 \times f^{\prime\#}(z) = 0$. If we write $f(z) = u(z) + iv(z)$ it follows, moreover, that $u(z)$ and $v(z)$ are both \#-continuous.

**Remark 15.1.** When we consider the \#-derivative of a $\mathbb{C}_c^\#$-valued function, defined on a set $A \subset \mathbb{C}_c^\#$ in the complex plane $\mathbb{C}_c^\#$, it is of course understood that $z \in A$ and that the limit is with respect to values $h$ such that $z+h \in A$. The existence of the \#-derivative will therefore have a different meaning depending on whether $z$ is an interior point or a \#-boundary point of $A$. The way to avoid this is to insist that all \#-analytic functions be defined on open sets.

**Definition 15.1.** A $\mathbb{C}_c^\#$-valued function $f(z)$, defined on an open set $\Omega$, is said to be $\mathbb{C}_c^\#$-analytic in $\Omega$ if it has a \#-derivative at each point of $\Omega$. And more explicitly that $f(z)$ is \#-analytic function. A commonly used synonym is \#-holomorphic function.

**Definition 15.2.** A function $f(z)$ is \#-analytic on an arbitrary point set $A$ if it is the restriction to $A$ of a function which is \#-analytic in some open set containing $A$.

**Remark 15.2.** Note that the real and imaginary parts of an \#-analytic function in $\Omega$ satisfy the generalized Cauchy-Riemann equations

$$ \frac{\partial^{\#} u}{\partial^{\#} x} = \frac{\partial^{\#} v}{\partial^{\#} y}; \quad \frac{\partial^{\#} u}{\partial^{\#} y} = -\frac{\partial^{\#} v}{\partial^{\#} x}. \qquad (15.2) $$
Conversely, if $u$ and $v$ satisfy these equations in $\Omega$, and if the partial $\#$-derivatives are
$\#$-continuous, then $u + iv$ is an $\#$-analytic function in $\Omega$.

**Theorem 15.1.** An $\#$-analytic function $f$ in a region $\Omega$ whose $\#$-derivative vanishes identically must reduce to a constant. The same is true if either the real part, the imaginary part, the modulus, or the argument is constant.

15.2. The $C_c^\#$-valued $\#$-Exponential $\mathrm{Ext}\text{-exp}(z)$.

We define the $\#$-exponential $\mathrm{Ext}\text{-exp}(z)$ function as the solution of the differential equation

$$
f^{\prime \#}(z) = f(z), f(0) = 1. \tag{15.3}
$$

We solve it by setting

$$
f(z) = \mathrm{Ext} - \sum_{n=0}^{\infty} a_n z^n, f^{\prime \#}(z) = \mathrm{Ext} - \sum_{n=0}^{\infty} n a_n z^n. \quad (15.4)
$$

If Eq.(15.4) is to be satisfied, we must have $a_{n-1} = na_n, n \in \mathbb{N}^\#$ and the initial condition gives $a_0 = 1$. It follows by hypee infinite induction that $a_n = 1/n!$.

**Abbreviation 14.1.** The solution of the Eq.(15.4) is denoted by $\mathrm{Ext}\text{-}e^z$ or $\mathrm{Ext}\text{-}\exp(z)$ or $\mathrm{Ext}\text{-}\exp z$. Thus finally we obtain

$$
\mathrm{Ext}\text{-}\exp(z) = \mathrm{Ext} - \sum_{n=0}^{\infty} \frac{z^n}{n!}. \qquad (15.5)
$$

15.3. The $C_c^\#$-valued Trigonometric Functions $Ext\text{-sin}(z)$, $Ext\text{-cos}(z)$.

The $C_c^\#$-valued trigonometric functions $Ext\text{-sin}(z)$, $Ext\text{-cos}(z)$ are defined by

$$
Ext\text{-}\sin(z) = \frac{1}{2}(Ext\text{-}\exp(iz) - Ext\text{-}\exp(-iz)) \quad (15.6)
$$

and

$$
Ext\text{-}\cos(z) = \frac{1}{2}(Ext\text{-}\exp(iz) + Ext\text{-}\exp(-iz)). \quad (15.7)
$$

Substitution (14.)-(14.) in (14.) gives that

$$
Ext\text{-}\sin(z) = \tag{15.8}
$$

and

$$
Ext\text{-}\cos(z) = \tag{15.9}
$$

From (14) we obtain generalized Euler’s formula

$$
\mathrm{Ext}\text{-}\exp(iz) = \mathrm{Ext}\text{-}\cos(z) + i(\mathrm{Ext}\text{-}\sin(z)) \quad (15.10)
$$

and as well as the identity

$$
(\mathrm{Ext}\text{-}\sin(z))^2 + (\mathrm{Ext}\text{-}\cos(z))^2 = 1. \qquad (15.11)
$$

15.4. The periodicity of the $\#$-exponential $\mathrm{Ext}\text{-exp}(iz)$.

**Definition 15.4.** We say that $f(z)$ has the period $c$ if $f(z+c) = f(z)$ for all $z \in C_c^\#$.

Thus a period of $Ext\text{-}e^z$ satisfies $Ext\text{-}e^{z+c} = Ext\text{-}e^z$, or $Ext\text{-}e^c = 1$. It follows that $c = i\omega$
with real $\omega \in \mathbb{R}_c^\#$ we prefer to say that $\omega$ is a period of $Ext\text{-}e^{iz}$. We shall show that there
are periods, and that they are all integral multiples of a positive period $\omega_0$. From
$(Ext-\sin(y))'^{\#} = Ext-\cos(y) \le 1$ and $Ext-\sin(0) = 0$ one obtains $Ext-\sin(y) < y$ for $y > 0$, either by integration or by use of the generalized mean-value theorem. In the same way $(Ext-\cos(y))'^{\#} = -Ext-\sin(y) > -y$ and $Ext-\cos(0) = 1$ gives $Ext-\cos(y) > 1 - y^2/2$, which in turn leads to $Ext-\sin(y) > y - y^3/6$ and finally to $Ext-\cos(y) < 1 - y^2/2 + y^4/24$. This inequality shows that $Ext-\cos(\sqrt{3}) < 0$, and therefore there is a $y_0$ such that $0 < y_0 < \sqrt{3}$ and $Ext-\cos(y_0) = 0$. Because $(Ext-\sin(y_0))^2 + (Ext-\cos(y_0))^2 = 1$, we have $Ext-\sin(y_0) = \pm 1$, that is, $Ext-e^{iy_0} = \pm i$, and hence $Ext-e^{4iy_0} = 1$. We have shown that $4y_0$ is a period. Actually, it is the smallest positive period. To see this, take $0 < y < y_0$.

Then $Ext-\sin(y) > y(1 - y^2/6) > y/2 > 0$, which shows that $Ext-\cos(y)$ is strictly decreasing. Because $Ext-\sin(y)$ is positive and $(Ext-\sin(y))^2 + (Ext-\cos(y))^2 = 1$ it follows that $Ext-\sin(y)$ is strictly increasing, and hence $Ext-\sin(y) < Ext-\sin(y_0) = 1$.

The double inequality $0 < Ext-\sin(y) < 1$ guarantees that $Ext-e^{iy}$ is neither $\pm 1$ nor $\pm i$. Therefore $Ext-e^{4iy} \ne 1$, and $4y_0$ is indeed the smallest positive period. We denote it by $\omega_0$. Consider now an arbitrary period $\omega_0$. There exists an integer $n$ such that $n\omega_0 \le \omega < (n + 1)\omega_0$. If $w$ were not equal to $n\omega_0$, then $\omega - n\omega_0$ would be a positive period $< \omega_0$. Since this is not possible, every period must be an integral multiple of $\omega_0$.

**Abbreviation 15.2.** The smallest positive period of $Ext-e^{iz}$ is denoted by $2\pi_{\#}$.

**Remark 15.3.** Note that $st(\pi_{\#}) = \pi \in \mathbb{R}$.

## 15.5. The $\mathbb{C}_c^\#$-valued Logarithm.

Together with the exponential function $Ext-e^{iz}$ we must also introduce its inverse function, the $\mathbb{C}_c^\#$-valued logarithm. By definition, $z = Ext-\log w$ is a root of the equation $Ext-e^{iz} = w$. First of all, since $Ext-e^{iz}$ is always $\ne 0$, the number 0 has no logarithm. For $w \ne 0$ the equation $Ext-e^{x+iy} = w$ is equivalent to

$$Ext-e^{iz} = |w|, Ext-e^{iy} = \frac{w}{|w|}. \qquad (15.12)$$

The first equation has a unique solution $x = Ext-\log|w|$, the $\mathbb{R}_c^\#$-valued logarithm of the positive number $|w| \in \mathbb{R}_c^\#$. The right-hand member of the second equation (15.12) is a complex number in $\mathbb{C}_c^\#$ of absolute value 1. Therefore, as we have just seen, it has one and only one solution in the interval $0 \le y < 2\pi_{\#}$. In addition, it is also satisfied by all $y$ that differ from this solution by an integral multiple of $2\pi_{\#}$. We see that every complex number other than 0 has hyper infinitely many logarithms which differ from each other by multiples of $2\pi_{\#}i$.

The imaginary part of $Ext-\log w$ is also called the argument of $w$, $Ext-\arg w$, and it is interpreted geometrically as the angle, measured in radians, between the positive real axis and the half line from 0 through the point $w$. According to this definition the argument has hyper infinitely many values which differ by multiples of $2\pi_{\#}$, and

$$Ext-\log w = Ext-\log|w| + i \arg w. \qquad (15.13)$$

**Remark 15.4.** The addition property of the exponential function $Ext-e^{iz}$ implies

$$Ext-\log(z_1 \times z_2) = Ext-\log z_1 + Ext-\log z_2, \qquad (15.14) \\ Ext-\arg(z_1 \times z_2) = Ext-\arg z_1 + Ext-\arg z_2,$$

but only in the sense that both sides represent the same hyper infinite set of
complex numbers. The inverse of $Ext\cos(z)$ is obtained by solving the equation

$$Ext\cos(z) = \frac{1}{2}(Exte^{iz} + Ext\bar{e}^{-iz}) = w. \qquad (15.15)$$

This is a quadratic equation in $Ext\cos(z)$ with the roots

$$Ext\cos(z) = w \pm \sqrt{w^2 - 1} \qquad (15.16)$$

and therefore

$$z = Ext\arccos(w) = -i(Ext\log(w \pm \sqrt{w^2+1})), \qquad (15.17)$$

or in the form

$$Ext\arccos(w) = \pm i(Ext\log(w + \sqrt{w^2+1})), \qquad (15.18)$$

The hyper infinitely many values of $Ext\arccos(w)$ reflect the evenness and periodicity of $Ext\cos(w)$. The inverse sine is most easily defined by formula

$$Ext\arcsin(w) = \frac{\pi}{2} - (Ext\arccos(w)). \qquad (15.19)$$

# 16. Complex Integration of the $C_c^\#$-valued function $f(t)$.

## 16.1. Definition and basic properties of the complex integral.

If $f(t) = u(t) + iv(t)$ is a $\#$-continuous function, defined in an interval $(a,b)$, we set by definition

$$Ext\int_a^b f(t)d^\#t = Ext\int_a^b u(t)d^\#t + i\left(Ext\int_a^b v(t)d^\#t\right). \qquad (16.1)$$

This integral has most of the properties of the real integral. In particular, if $c = \alpha + i\beta$ is a complex constant we obtain

$$Ext\int_a^b cf(t)d^\#t = c\left(Ext\int_a^b f(t)d^\#t\right). \qquad (16.2)$$

The fundamental inequality

$$\left|Ext\int_a^b f(t)d^\#t\right| \leq Ext\int_a^b |f(t)|d^\#t. \qquad (16.3)$$

holds for arbitrary $C_c^\#$-valued function $f(t)$.

We consider now a piecewise $\#$-differentiable arc $\gamma$ with the equation

$$z = z(t), a \leq t \leq b.$$

If the function $f(z)$ is defined and $\#$-continuous on $\gamma$, then $f(z(t))$ is also $\#$-continuous and we can set

$$v \quad Ext\int_{\gamma} f(z)d^\#z = Ext\int_{a}^{b} f(z(t))z^{\prime\#}(t)d^\#t. \qquad (16.4)$$

The most important property of the integral (16.4) is its invariance under a change of parameter. A change of parameter is determined by an increasing function $t = t(\tau)$ which maps an interval $\alpha \leq \tau \leq \beta$ onto $a \leq t \leq b$; we assume that $t(\tau)$ is piecewise
#-differentiable. By the rule for changing the variable of integration we get

$$Ext- \int_{a}^{b} f(z(t))z'^{\#}(t)d^{\#}t = Ext- \int_{a}^{\beta} f(z(t(\tau)))z'^{\#}(t(\tau))t'^{\#}(\tau)d^{\#}\tau. \quad (16.5)$$

We defined the opposite arc-$\gamma$ by the equation $z = z(-t)$, $-b \le t \le -a$. We have thus

$$Ext- \int_{-\gamma} f(z)d^{\#}z = - \left( Ext- \int_{\gamma} f(z)d^{\#}z \right). \quad (16.6)$$

The integral (16.4) has also a very obvious additive property. It is clear what is meant by subdividing an arc $\gamma$ into a finite or hyperfinite number of subarcs. A subdivision can be indicated by a symbolic equation: $\gamma = \gamma_1 + \gamma_2 + \dots + \gamma_n, n \in \mathbb{N}^\#$, and the corresponding integrals satisfy the relation

$$Ext- \int_{\gamma_1+\gamma_2+\dots+\gamma_n} f(z)d^{\#}z = Ext- \sum_{i=1}^{n} \left( Ext- \int_{\gamma_i} f(z)d^{\#}z \right). \quad (16.7)$$

Finally, the integral over a closed curve is also invariant under a shift of parameter. The old and the new initial point determine two subarcs $\gamma_1, \gamma_2$, and the invariance follows from the fact that the integral over $\gamma_1 + \gamma_2$ is equal to the integral over $\gamma_2 + \gamma_1$. In addition to integrals of the form (16.4) we can also consider line integrals with respect to $\bar{z}$. The most convenient definition is by double conjugation

$$Ext- \int_{\gamma} f(z) \overline{d^{\#}z} = Ext- \int_{\gamma} f(z) d^{\#}z. \quad (16.8)$$

Using notation (16.7), line integrals with respect to $x$ or $y$ can be introduced by

$$Ext- \int_{\gamma} f(z) d^{\#}x = \frac{1}{2} \left( Ext- \int_{\gamma} f(z) d^{\#}z + Ext- \int_{\gamma} f(z) \overline{d^{\#}z} \right), \quad (16.9) \\ Ext- \int_{\gamma} f(z) d^{\#}y = \frac{1}{2i} \left( Ext- \int_{\gamma} f(z) d^{\#}z - Ext- \int_{\gamma} f(z) \overline{d^{\#}z} \right).$$

With $f = u + iv$ we find that the integral (16.4) can be written in the form

$$Ext- \int_{\gamma} (ud^{\#}x - vd^{\#}y) + i \left( Ext- \int_{\gamma} (ud^{\#}y + vd^{\#}x) \right). \quad (16.10)$$

Of course we could just as well have started by defining integrals of the form

$$Ext- \int_{\gamma} (pd^{\#}x + qd^{\#}y), \quad (16.11)$$

in which case formula (16.10) would serve as definition of the integral (16.4). An essentially different line integral is obtained by integration with respect to arc length. Two notations are in common use, and the definition is
$$ \mathrm{Ext}- \int_{\gamma} f d^{\#} s = \mathrm{Ext}- \int_{\gamma} f(z) |d^{\#} z| = \mathrm{Ext}- \int_{\gamma} f(z(t)) |z'^{\#}(t)| d^{\#} t. \qquad (16.12) $$

This integral is again independent of the choice of parameter. In contrast to (16.6) we get

$$ \mathrm{Ext}- \int_{-\gamma} f(z) |d^{\#} z| = \mathrm{Ext}- \int_{\gamma} f(z) |d^{\#} z|, \qquad (16.13) $$

while (16.7) remains valid in the same form. The inequality

$$ \left| \mathrm{Ext}- \int_{\gamma} f(z) d^{\#} z \right| \leq \mathrm{Ext}- \int_{\gamma} |f(z)| |d^{\#} z| \qquad (16.14) $$

is a consequence of (16.3).

**Remark 16.1.** For $f \equiv 1$ the integral (16.12) reduces to $\int_{\gamma} |dz|$ which is by definition the length of $\gamma$. As an example we compute the length of a circle. From the parametric equation $z = z(t) = a + \rho(\mathrm{Ext}-e^{it})$, $0 \le t \le 2\pi_{\#}r$, of a full circle we obtain $z'^{\#}(t) = i\rho(\mathrm{Ext}-e^{it})$ and hence

$$ \int_{0}^{2\pi_{\#}} |z'^{\#}(t)| d^{\#} t = \int_{0}^{2\pi_{\#}} \rho d^{\#} t = 2\pi_{\#} \rho \qquad (16.15) $$

as expected.

## 16.2. Line Integrals as Functions of Arcs.

Remind that the length of an arc can also be defined as the least upper bound of all hyperfinite sums

$$ \mathrm{Ext}- \sum_{i=1}^{n} |z(t_i) - z(t_{i-1})|, \qquad (16.16) $$

$n \in \mathbb{N}^{\#}/\mathbb{N}$, where $a = t_0 < t_1 < \dots < t_n = b$. If this least upper bound is finite or hyperfinite we say that the arc is rectifiable. It is quite easy to show that piecewise \#-differentiable arcs are rectifiable, and that the two definitions of length coincide. It is clear that the sums (16.6) and the corresponding sums

$$ \mathrm{Ext}- \sum_{i=1}^{n} |x(t_i) - x(t_{i-1})|; \mathrm{Ext}- \sum_{i=1}^{n} |y(t_i) - y(t_{i-1})|, \qquad (16.17) $$

where $z(t) = x(t) + iy(t)$, are bounded or hyperbounded at the same time. When the latter sums are bounded (or hyperbounded), one says that the functions $x(t)$ and $y(t)$ are of bounded (or hyperbounded) variation. An arc $z = z(t)$ is rectifiable if and only if the real and imaginary parts of $z(t)$ are of bounded (or hyperbounded) variation. If $\gamma$ is rectifiable and $f(z)$ \#-continuous on $\gamma$ it is possible to define integrals of type (16.12) as a \#-limit

$$ \mathrm{Ext}- \int_{\gamma} f d^{\#} s = \#\lim_{n \to \infty} \left( \mathrm{Ext}- \sum_{k=1}^{n} f(z(t_k)) |z(t_i) - z(t_{i-1})| \right). \qquad (16.18) $$

General line integral of the form $\mathrm{Ext}- \int_{\gamma} (pd^{\#}x + qd^{\#}y)$ can be considered as functional
of the arc $\gamma$. It is then assumed that $p$ and $q$ are defined and $\#$-continuous in a region $\Omega$ and that $\gamma$ is free to vary in $\Omega$. An important class of integrals is characterized by the property that the integral over an arc depends only on its end points. In other words, if $\gamma_1$ and $\gamma_2$ have the same initial point and the same end point, we require that

$$ \mathrm{Ext}-\int_{\gamma_1} (pd^\#x + qd^\#y) = \mathrm{Ext}-\int_{\gamma_2} (pd^\#x + qd^\#y). \quad (16.19) $$

To say that an integral depends only on the end points is equivalent to saying that the integral over any closed curve is zero. Indeed, if $\gamma$ is a closed curve, then $\gamma$ and $-\gamma$ have the same end points, and if the integral depends only on the end points, we obtain

$$ \mathrm{Ext}-\int_{\gamma} (pd^\#x + qd^\#y) = \mathrm{Ext}-\int_{-\gamma} (pd^\#x + qd^\#y) = -\left( \mathrm{Ext}-\int_{\gamma} (pd^\#x + qd^\#y) \right) \quad (16.20) $$

and consequently $\int_{\gamma} (pd^\#x + qd^\#y) = 0$. Conversely, if $\gamma_1$ and $\gamma_2$ have the same end points, then $\gamma_1 - \gamma_2$ is a closed curve, and if the integral over any closed curve vanishes, it follows that $\mathrm{Ext}-\int_{\gamma_1} (pd^\#x + qd^\#y) = \mathrm{Ext}-\int_{\gamma_2} (pd^\#x + qd^\#y)$.

<figure><img src="image_2.png" /><figcaption>Pic. 1.</figcaption></figure>

The following theorem gives a necessary and sufficient condition under which a line integral depends only on the end points.

**Theorem 16.1.** The line integral $\mathrm{Ext}-\int_{\gamma} (pd^\#x + qd^\#y)$, defined in $\Omega$, depends only on the end points of $\gamma$ if and only if there exists a function $U(x, y)$ in $\Omega$ with the partial $\#$-derivatives $\partial^\#u/\partial^\#x = p, \partial^\#u/\partial^\#y = q$.

The sufficiency follows at once, for if the condition is fulfilled we can write, with the usual notations,

$$ \begin{aligned} \mathrm{Ext}-\int_{\gamma} (pd^\#x + qd^\#y) &= \mathrm{Ext}-\int_{a}^{b} \left[ \frac{\partial^\# U}{\partial^\# x} x'^\#(t) + \frac{\partial^\# U}{\partial^\# y} y'^\#(t) \right] d^\#t = \\ \mathrm{Ext}-\int_{a}^{b} \frac{d^\# t}{d^\# t} U(x(t), y(t)) d^\#t &= U(x(b), y(b)) - U(x(a), y(a)). \end{aligned} \quad (16.21) $$

and the value of this difference depends only on the end points. To prove the necessity we choose a fixed point $(x_0, y_0) \in \Omega$, join it to $(x, y)$ by a polygon $\gamma$,
contained in Ω, whose sides are parallel to the coordinate axes (Pic.1) and define
a function $U(x, y)$ by

$$
U(x, y) = \operatorname{Ext-} \int_{\gamma} (pd^{\#}x + qd^{\#}y). \qquad (16.22)
$$

Since the integral depends only on the end points, the function is well defined.
Moreover, if we choose the last segment of $\gamma$ horizontal, we can keep $y$ constant and
let $x$ vary without changing the other segments. On the last segment we can choose $x$
for parameter and obtain

$$
U(x, y) = \operatorname{Ext-} \int_{y}^{x} p(x, y) d^{\#}x + \operatorname{const.}, \qquad (16.23)
$$

the lower limit of the integral being irrelevant. From Eq.(16.23) it follows at once that

$$
\frac{\partial^{\#} U}{\partial^{\#} x} = p. \text{ In the same way, by choosing the last segment vertical, we can show that}
$$

$$
\frac{\partial^{\#} U}{\partial^{\#} y} = q. \text{ It is customary to write } d^{\#} U = (\partial^{\#} U / \partial^{\#} x) d^{\#} x + (\partial^{\#} U / \partial^{\#} y) d^{\#} y \text{ and to say that}
$$

an expression $pd^{\#}x + qd^{\#}y$ which can be written in this form is an exact #-differential.
Thus an integral depends only on the end points if and only if the integrand is an
exact differential. Observe that $p, q$ and $U$ can be either real or complex. The function
$U$, if it exists, is uniquely determined up to an additive constant, for if two functions
have the same partial #-derivatives their #-difference must be constant.

When is $f(z)d^\#z = f(z)d^\#x + if(z)d^\#y$ an exact #-differential? According to the definition
there must exist a function $F(z)$ in $\Omega$ with the partial #-derivatives

$$
\frac{\partial^{\#} F(z)}{\partial^{\#} x} = f(z), \quad \frac{\partial^{\#} F(z)}{\partial^{\#} y} = if(z). \qquad (16.24)
$$

If this is so, $F(z)$ fulfills the generalized Cauchy-Riemann equation

$$
\frac{\partial^{\#} F(z)}{\partial^{\#} x} = i \frac{\partial^{\#} F(z)}{\partial^{\#} y}, \qquad (16.25)
$$

since $f(z)$ is by assumption #-continuous $F(z)$ is #-analytic with the #-derivative $f(z)$.
The integral $\operatorname{Ext-}\int_{\gamma} fd^{\#}z$, with #-continuous $f$, depends only on the end points of $\gamma$ if
and only iff is the derivative of an analytic function in $\Omega$. Under these circumstances
we shall prove later that $f(z)$ is itself #-analytic.

As an immediate application of the above result we find that

$$
\int_{\gamma} (z-a)^n d^\# z = 0 \qquad (16.26)
$$

for all closed curves $\gamma$, provided that the integer $n \in \mathbb{N}^\#$ is $\ge 0$. In fact, $(z-a)^n$ is the
\#-derivative of $(z-a)^{n+1}/(n+1)$, a function which is \#-analytic in the whole plane $\mathbb{C}^\#$.
If $n$ is negative, but $\ne -1$, the same result holds for all closed curves which do not
pass through $a$, for in the complementary region of the point $a$ the indefinite integral
is still \#-analytic and single-valued. For $n = -1$, Eq.(16.26) does not always hold.
Consider a circle $C$ with the center $a$, represented by the equation $z = a + \rho(\operatorname{Ext}-e^{it})$,
$0 \le t \le 2\pi$. We obtain
$$ \int_{\gamma} \frac{d^{\#}z}{(z-a)} = \int_{0}^{2\pi_{\#}} id^{\#}t = 2\pi_{\#}i. \qquad (16.27) $$

This result shows that it is impossible to define a single-valued branch of $Ext\text{-log}(z-a)$ in an annulus $\rho_1 < |z-a| < \rho_2$. On the other hand, if the closed curve $\gamma$ is contained in a half plane which does not contain $a$, the integral vanishes, for in such a half plane a single-valued and \#-analytic branch of $Ext\text{-log}(z-a)$ can be defined.

## 16.3. Generalized Cauchy's Theorem for a Rectangle.

We consider, specifically, a rectangle $R \subset \mathbb{C}^\#$ defined by inequalities $a \le x \le b$, $c \le y \le d$. Its perimeter can be considered as a simple closed curve consisting of four line segments whose direction we choose so that $R$ lies to the left of the directed segments. The order of the vertices is thus $(a, c)$, $(b, c)$, $(b, d)$, $(a, d)$. We refer to this closed curve as the boundary curve or contour of $R$, and we denote it by $\partial^\# R$

<figure><img src="image_3.png" /><figcaption>Pic.2. Bisection of rectangle.</figcaption></figure>

**Theorem 16.2.** If the function $f(z)$ is \#-analytic on $R$, then

$$ \operatorname{Ext-} \int_{\partial^{\#} R} f(z) d^{\#} z = 0. \qquad (16.28) $$

**Proof.** The proof is based on the method of bisection. Let us introduce the notation

$$ \eta(R) = \operatorname{Ext-} \int_{\partial^{\#} R} f(z) d^{\#} z. \qquad (16.29) $$

If $R$ is divided into four congruent rectangles $R^{(1)}, R^{(2)}, R^{(3)}, R^{(4)}$, we get

$$ \eta(R) = \eta(R^{(1)}) + \eta(R^{(2)}) + \eta(R^{(3)}) + \eta(R^{(4)}). \qquad (16.30) $$

for the integrals over the common sides cancel each other, see Pic.1. It follows from Eq.(16.) that at least one of the rectangles $R^{(k)}$, $k = 1, 2, 3, 4$, must satisfy the condition $|\eta(R^{(k)})| \ge |\eta(R)|/4$. This process can be repeated inductively by hyper infinite induction, and we obtain a hyper infinite sequence of nested rectangles $R \supset R_1 \supset R_2 \dots \supset R_n \dots \supset \dots$ with the property $|\eta(R_n)| \ge 4^{-n}|\eta(R_{n-1})|$, $n \in \mathbb{N}^\#$. Thus

$$ |\eta(R_n)| \ge 4^{-n}|\eta(R)|. \qquad (16.31) $$

The rectangles $R_n$ converge to a point $z^* \in R$ in the sense that $R_n$ will be contained in a prescribed neighborhood $|z - z^*| < \delta$ as soon as $n \in \mathbb{N}^\# \mathbb{N}$ is sufficiently large.
First of all, we choose $\delta$ so small that $f(z)$ is defined and $\#$-analytic in $|z - z^*| < \delta$, $\delta \approx 0$. Secondly, if $\varepsilon > 0$, $\varepsilon \approx 0$ is given, we can choose $\delta$ such that

$$ \left| \frac{f(z) - f(z^*)}{z - z^*} - f^{\prime\#}(z^*) \right| < \varepsilon, \qquad (16.32) $$

and therefore

$$ |f(z) - f(z^*) - (z - z^*) f^{\prime\#}(z^*)| < \varepsilon |z - z^*|. \qquad (16.33) $$

for $|z - z^*| < \delta$. We assume that $\delta$ satisfies both conditions and that $R_n$ is contained in $|z - z^*| < \delta$. We make now the observation that

$$ \mathrm{Ext} - \int_{\partial^{\#} R_n} d^{\#} z = 0, \mathrm{Ext} - \int_{\partial^{\#} R_n} zd^{\#} z = 0 \qquad (16.34) $$

By virtue of the equations (16.34) we are able to write

$$ |\eta(R_n)| = \mathrm{Ext} - \int_{\partial^{\#} R_n} |f(z) - f(z^*) - (z - z^*) f^{\prime\#}(z^*)| d^{\#} z \qquad (16.35) $$

and it follows by (16.33) that

$$ |\eta(R_n)| \le \varepsilon \left( \mathrm{Ext} - \int_{\partial^{\#} R_n} |z - z^*| \times |d^{\#} z| \right). \qquad (16.36) $$

In the last integral $|z - z^*|$ is at most equal to the length $d_n$ of the diagonal of $R_n$. If $L_n$ denotes the length of the perimeter of $R_n$, the integral is hence $\le d_n L_n$. But if $d$ and $L$ are the corresponding quantities for the original rectangle $R$, it is clear that $d_n = 2^{-n}d$ and $L_n = 2^{-n}L$. By (16.36) we have hence

$$ |\eta(R_n)| \le 4^{-n} dL\varepsilon \qquad (16.37) $$

and comparison with (16.31) yields

$$ |\eta(R)| \le dL\varepsilon. \qquad (16.38) $$

Since $\varepsilon \approx 0$ is arbitrary, we can only have $\eta(R) \equiv 0$, and the theorem is proved. **Theorem 16.3.** Let $f(z)$ be $\#$-analytic on the set $R'$ obtained from a rectangle $R$ by omitting a finite or hyperfinite number of interior points $\zeta_j$. If it is true that $\#\lim_{z \to \#} \zeta_j(z - \zeta_j)f(z) = 0$ for all $j \in \mathbb{N}^\#$, then $\mathrm{Ext} - \int_{\partial^{\#} R} f(z) d^{\#} z = 0$.

**Proof.** It is sufficient to consider the case of a single exceptional point $\zeta$, for evidently $R$ can be divided into smaller rectangles which contain at most one $\zeta_j$. We divide now $R$ into nine rectangles, as shown in Pic.2, and apply Theorem 16.2 to all but the rectangle $R_0$ in the center. If the corresponding equations (16.28) are added, we obtain,

after cancellations,

$$ \mathrm{Ext} - \int_{\partial^{\#} R} f(z) d^{\#} z = \mathrm{Ext} - \int_{\partial^{\#} R_0} f(z) d^{\#} z \qquad (16.39) $$
<figure><img src="image_4.png" /><figcaption>Pic. 3.</figcaption></figure>

If $\varepsilon > 0, \varepsilon \approx 0$ we can choose the rectangle $R_0$ so infinite small that $|f(z)| \le \varepsilon|z - \zeta|$ on $\partial^{\#}R_0$. By (16.39) we have thus

$$ \left| \mathrm{Ext} - \int_{\partial^{\#}R} f(z) d^{\#}z \right| = \varepsilon \left( \mathrm{Ext} - \int_{\partial^{\#}R_0} \frac{|d^{\#}z|}{|z - \zeta|} \right) \quad (16.40) $$

If we assume, as we may, that $R_0$ is a square of center $\zeta$, elementary estimates show that

$$ \mathrm{Ext} - \int_{\partial^{\#}R_0} \frac{|d^{\#}z|}{|z - \zeta|} < 8. \quad (16.41) $$

Thus finally we obtain

$$ \left| \mathrm{Ext} - \int_{\partial^{\#}R} f(z) d^{\#}z \right| < 8\varepsilon. \quad (16.42) $$

and since $\varepsilon$ is arbitrary the theorem follows. We conclude that the hypothesis of the theorem is certainly fulfilled if $f(z)$ is $\#$-analytic and bounded or hyperbounded on $R'$.

## 16.4. Generalized Cauchy's Theorem in a Disk.

It is not true that the integral of an $\#$-analytic function over a closed curve is always zero. For example

$$ \int_C \frac{d^{\#}z}{|z-a|} = 2i\pi_. \quad (16.43) $$

**Theorem 16.4.** If $f(z)$ is $\#$-analytic in an open disk $\Delta$, then

$$ \mathrm{Ext} - \int_{\gamma} f(z) d^{\#}z = 0 \quad (16.44) $$

for every closed curve $\gamma \subset \Delta$.

**Proof.** We define a function $F(z)$ by

$$ F(z) = \mathrm{Ext} - \int_{\sigma} f(z) d^{\#}z, \quad (16.45) $$

where $\sigma$ consists of the horizontal line segment from the center $(x_0, y_0)$ to $(x, y_0)$ and the vertical segment from $(x, y_0)$ to $(x, y)$; it is immediately seen that $\partial^{\#}F/\partial^{\#}y = if(z)$. On
the other hand, by Theorem 16.2 $\sigma$ can be replaced by a path consisting of a vertical segment followed by a horizontal segment. This choice defines the same function $F(z)$, and we obtain $\partial^\# F/\partial^\# x = f(z)$. Hence $F(z)$ is #-analytic in $\Delta$. with the #-derivative $f(z)$, and $f(z)d^\#z$ is an exact #-differential.

**Theorem 16.5.** Let $f(z)$ be #-analytic in the region $\Delta'$ obtained by omitting a finite or hyperfinite number of points $\zeta_j$ from an open disk $\Delta$. If $f(z)$ satisfies the condition $\#\lim_{z\to\#} \zeta_j(z-\zeta_j)f(z) = 0$ for all $j$, then (16.44) holds for any closed curve $\gamma \subset \Delta'$.

<figure><img src="image_5.png" /><figcaption>Pic.4.</figcaption></figure>

The proof must be modified, for we cannot let $rr$ pass through the exceptional points. Assume first that no $\zeta_j$ lies on the lines $x = x_0$ and $y = y_0$. It is then possible to avoid the exceptional points by letting $\sigma$ consist of three segments (Pic.4). By an obvious application of Theorem 16.3 we find that the value of $F(z)$ in (16.44) is independent of the choice of the middle segment; moreover, the last segment can be either vertical or horizontal. We conclude as before that $F(z)$ is an indefinite integral of $f(z)$, and the theorem follows.

## 16.5. Generalized Cauchy's integral formula.

Through a very simple application of the generalized Cauchy's theorem it becomes possible to represent an #-analytic function $f(z)$ as a line integral in which the variable $z \in \mathbb{R}_c^\#$ enters as a parameter. This representation, known in classical case as Cauchy's integral formula, has numerous important applications. Above all, it enables us to study the local properties of an #-analytic function in full detail.

**Lemma 16.1.** If the piecewise #-differentiable closed curve $\gamma$ does not pass through the point $a$, then the value of the integral

$$ \int_{\gamma} \frac{d^\# z}{|z-a|}. \qquad (16.46) $$

is a multiple of $2i\pi\#_a$.

**Definition 16.1.** We define the index of the point $a$ with respect to the curve $\gamma$ by the equation

$$ n(\gamma, a) = \frac{1}{2\pi i} \int_{\gamma} \frac{d^\# z}{z-a}. \qquad (16.47) $$

The index (16.47) is also called the winding number of $\gamma$ with respect to $a$. It is clear that $n(-\gamma, a) = -n(\gamma, a)$. The following property is an immediate consequence of
**Theorem 16.4.**

(i) If $\gamma$ lies inside of a circle, then $n(\gamma, a) = 0$ for all points outside of the same circle. As a point set $\gamma$ is #-closed and bounded (or hyperbounded). Its complement is #-open and can be represented as a union of disjoint regions, the components of the complement. We shall say, for short, that $\gamma$ determines these regions. If the complementary regions are considered in the extended plane, there is exactly one which contains the point at infinity. Consequently, $\gamma$ determines one and only one unbounded region.

(ii) As a function of a the index $n(\gamma, a)$ is constant in each of the regions determined by $\gamma$, and zero in the unbounded region.

Any two points in the same region determined by $\gamma$ can be joined by a polygon which does not meet $\gamma$. For this reason it is sufficient to prove that $n(\gamma, a) = n(\gamma, b)$ if $\gamma$ does not meet the line segment from $a$ to $b$. Outside of this segment the function $(z-a)/(z-b)$ is never real and $\le 0$. For this reason the principal branch of $\text{Ext}\text{-log}[(z-a)/(z-b)]$ is #-analytic in the complement of the segment. Its derivative is equal to $(z-a)^{-1} - (z-b)^{-1}$, and if $\gamma$ does not meet the segment we get

$$ \text{Ext}\text{-}\int_{\gamma} \left( \frac{1}{z-a} - \frac{1}{z-b} \right) d^\#z = 0; \qquad (16.48) $$

hence $n(\gamma, a) = n(\gamma, b)$. If $\text{lal}$ is sufficiently large, $\gamma$ is contained in a disk $|z| < \rho < |a|$ and we conclude by (i) that $n(\gamma, a) = 0$. This proves that $n(\gamma, a) = 0$ in the unbounded region.

We shall find the case $n(\gamma, a) = 1$ particularly important, and it is desirable to formulate a geometric condition which leads to this consequence.

For simplicity we take $a = 0$.

**Lemma 16.2.** Let $z_1, z_2$ be two points on a closed curve $\gamma$ which does not pass through the origin. Denote the subarc from $z_1$ to $z_2$ in the direction of the curve by $\gamma_1$, and the subarc from $z_2$ to $z_1$ by $\gamma_2$. Suppose that $z_1$ lies in the lower half plane and $z_2$ in the upper half plane. If $\gamma_1$ does not meet the negative real axis and $\gamma_2$ does not meet the positive real axis, then $n(\gamma, 0) = 1$.

For the proof we draw the half lines $L_1$ and $L_2$ from the origin through $z_1$ and $z_2$ (Pic. 4-5). Let $s_1, s_2$ be the points in which $L_1, L_2$ intersect a circle $C$ about the origin. If $C$ is described in the positive sense, the arc $C_1$ from $s_1$ to $s_2$ does not intersect the negative axis, and the arc $C_2$ from $s_2$ to $s_1$ does not intersect the positive axis. Denote the directed line segments from $z_1$ to $s_1$ and from $z_2$ to $s_2$ by $\delta_1, \delta_2$. Introducing the closed curves $\sigma_1 = \gamma_1 + \delta_2 - C_1 - \delta_1, \sigma_2 = \gamma_2 + \delta_1 - C_2 - \delta_2$ we get that $n(\gamma, 0) = n(C, 0) + n(\sigma_1, 0) + n(\sigma_2, 0)$ because of cancellations. But $\sigma_1$ does not meet the negative axis. Hence the origin belongs to the unbounded region determined by $\sigma_1$, and we obtain $n(\sigma_1, 0) = 0$. For a similar reason $n(\sigma_2, 0) = 0$, and we conclude that $n(\gamma, 0) = n(C, 0) = 1$.
<figure><img src="image_6.png" /><figcaption>Pic.5</figcaption></figure>

Let $f(z)$ be $\#$-analytic in an open disk $\Delta$. Consider a closed curve $\gamma \subset \Delta$. and a point $a \in \Delta$

which does not lie on $\gamma$. We apply Cauchy's theorem to the function

$$F(z) = \frac{f(z) - f(a)}{z - a}. \qquad (16.49)$$

This function is analytic for $z \neq a$. For $z = a$ it is not defined, but it satisfies the condition $\#\lim_{z \to a} [(z-a)F(z)] = \#\lim_{z \to a} [f(z) - f(a)] = 0$, which is the condition of Theorem 16.5. We conclude that

$$Ext - \int_{\gamma} \frac{f(z) - f(a)}{z - a} d^{\#}z = 0. \qquad (16.50)$$

This equation can be rewritten in the form

$$Ext - \int_{\gamma} \frac{f(z)d^{\#}z}{z-a} = f(a) \left( Ext - \int_{\gamma} \frac{d^{\#}z}{z-a} \right), \qquad (16.51)$$

and we observe that the integral in the right-hand member is by definition $2\pi i \text{in}(\gamma, a)$.

**Theorem 16.6.** Suppose that $f(z)$ is $\#$-analytic in an open disk $\Delta$, and let $\gamma$ be a closed curve in $\Delta$. For any point a not on $\gamma$

$$n(\gamma, a) \times f(a) = \frac{1}{2\pi i} \left( Ext - \int_{\gamma} \frac{f(z)d^{\#}z}{z-a} \right), \qquad (16.52)$$

where $n(\gamma, a)$ is the index of a with respect to $\gamma$.

In this statement we have suppressed the requirement that a be a point in $\Delta$. We have done so in view of the obvious interpretation of the formula (16.) for the case that a is not in $\Delta$. Indeed, in this case $n(\gamma, a)$ and the integral in the right-hand member are both zero.

It is clear that Theorem 6 remains valid for any region $\Omega$ to which Theorem 16.5 can be applied. The presence of exceptional points $\zeta_j$ is permitted, provided none of them coincides with $a$.

The most common application is to the case where $n(\gamma, a) = 1$. We have then
$$f(a) = \frac{1}{2\pi i} \left( \mathrm{Ext} - \int_{\gamma} \frac{f(z)d^\#z}{z-a} \right) \quad (16.)$$

and this we interpret as a representation formula. Indeed, it permits us to compute $f(a)$ as soon as the values of $f(z)$ on $\gamma$ are given, together with the fact that $f(z)$ is #-analytic in $\Delta$. In (16.) we may let a take different values, provided that the order of a with respect to $\gamma$ remains equal to 1. We may thus treat a as a variable, and it is convenient to change the notation and rewrite (16.) in the form

$$f(z) = \frac{1}{2\pi i} \left( \mathrm{Ext} - \int_{\gamma} \frac{f(\zeta)d^\#\zeta}{\zeta - z} \right) \quad (16.)$$

It is this formula which is usually referred to as Cauchy's integral formula. We must remember thflt it is valid only when $n(\gamma, z) = 1$, and that we have proved it only when $f(z)$ is #-analytic in a disk.

The representation formula (22) gives us a tool for the study of the local properties of #-analytic functions. In particular we can now show that an #-analytic function has #-derivatives of all orders $n \in \mathbb{N}^\#$, which are then also #-analytic.

We consider a function $f(z)$ which is #-analytic in an arbitrary region $\Omega$. To a point $a \in \Omega$ we determine a $\delta$-neighborhood $\Delta \subset \Omega$, and in $\Delta$ a circle $C$ about $a$. Theorem 6 can be applied to $f(z)$ in $\Delta$. Since $n(C, a) = 1$ we have $n(C, z) = 1$ for all points $z$ inside of $C$. For such $z$ we obtain by (22)

$$f(z) = \frac{1}{2\pi i} \left( \mathrm{Ext} - \int_C \frac{f(\zeta)d^\#\zeta}{\zeta - z} \right) \quad (16.)$$

Provided that the integral in(16.) can be #-differentiated under the sign of integration we find

$$f^{(n)}(z) = \frac{1}{2\pi i} \left( \mathrm{Ext} - \int_C \frac{f(\zeta)d^\#\zeta}{(\zeta - z)^n} \right) \quad (16.)$$

and

$$f^{(n)\#}(z) = \frac{1}{2\pi i} \left( \mathrm{Ext} - \int_C \frac{f(\zeta)d^\#\zeta}{(\zeta - z)^n} \right) \quad (16.)$$

If the #-differentiations can be justified, we shall have proved the existence of all #-derivatives at the points inside of $C$. Since every point in $\Omega$ lies inside of some such circle, the existence will be proved in the whole
region $\Omega$.

**Lemma 16.3.** Suppose that $\varphi(\zeta)$ is \#-continuous on the arc $\gamma$. Then the function

$$F_n(z) = \frac{1}{2\pi i} \left( \mathrm{Ext} - \int_{\gamma} \frac{\varphi(\zeta) d^\# \zeta}{(\zeta - z)^n} \right) \qquad (16.)$$

is analytic in each of the regions determined by $\gamma$, and its \#-derivative is $F_n^{\#}(z) = nF_{n+1}(z)$.

It is clear that Lemma 16.3 is just what is needed in order to deduce (23) and (24) in a rigorous way. We have thus proved that an analytic function has derivatives of all orders which are \#-analytic and can be represented by the formula (24).

**Theorem 16.7.** (Generalized Morera's theorem) If $f(z)$ is defined and \#-continuous in a region $\Omega$, and if $\mathrm{Ext} - \int_{\gamma} f(z) d^\# z = 0$ for all closed curves $\gamma$ in $\Omega$, then $f(z)$ is \#-analytic in $\Omega$.

## 16.6. Generalized Liouville's theorem.

**Theorem 16.8.** (Generalized Liouville's theorem) A function $f(z)$ which is \#-analytic and bounded in the whole plane $C_c^\#$ must reduce to a constant.

**Proof.** We make use of a simple estimate derived from (24). Let the radius of $C$ be $r$, and assume that $|f(z)| \le M$ on $C$. If we apply (24) with $z = a$, we obtain

$$|f^{(n)\#}(a)| \le Mn!r^{-n} \qquad (16.)$$

We need only the case $n = 1$. The hypothesis means that $|f(z)| \le M$ on all circles. Hence we can let $r$ tend to $\infty^\#$, and (25) leads to $f^{\#}(a) = 0$ for all $a$. We conclude that the function is constant.

## 16.7. Generalized fundamental theorem of algebra.

Liouville's theorem leads to an almost trivial proof of the generalized fundamental theorem of algebra.

**Theorem 16.9.** (Generalized fundamental theorem of algebra) Suppose that $P(z)$ is external polynomial of degree $n \in \mathbb{N}^\#$. The equation $P(z) = 0$ must have a root $\xi \in C_c^\#$.

**Proof.** Suppose that $P(z)$ is a polynomial of degree $n \in \mathbb{N}^\# \setminus \mathbb{N}$. If $P(z)$ were never zero, the function $1/P(z)$ would be \#-analytic in the whole plane $C_c^\#$. We know that $P(z) \to \infty^\#$, and therefore $1/P(z)$ tends to zero. This implies boundedness (the absolute value is \#-continuous on the Riemann sphere and has thus a finite or hyperfinite maximum), and by Liouville's theorem $1/P(z)$ would be constant. Since this is not so, the equation $P(z) = 0$ has a root.

# 17. The local properties of \#-analytic function.

## 17.1. Removable Singnlarities. Taylor's Theorem.

**Theorem 17.1.** Suppose that $f(z)$ is \#-analytic in the region $\Delta'$ obtained by omitting a point $a$ from a region $\Delta$. A necessary and sufficient condition that there exist an \#-analytic function in $\Delta$ which coincides with $f(z)$ in $\Delta'$ is that $\lim_{z \to a} (z - a)f(z) = 0$. The extended function is uniquely determined.

**Proof.** The necessity and the uniqueness are trivial since the extended function
must be #-continuous at a. To prove the sufficiency we draw a circle C about a so that C and its inside are contained in Δ. Cauchy's formula is valid, and therefore we have

$$f(z) = \frac{1}{2\pi i} \int_C \frac{f(\zeta)d^\#z}{\zeta - z} \qquad (17.1)$$

for all $z \neq a$ inside of C. But the integral in the right-hand member represents an #-analytic function of $z$ throughout the inside of C. Consequently, the function which is equal to $f(z)$ for $z \neq a$ and which has the value

$$\frac{1}{2\pi i} \int_C \frac{f(\zeta)d^\#z}{\zeta - z}. \qquad (17.2)$$

for $z = a$ is #-analytic in $\Delta$. It is natural to denote the extended function by $f(z)$ and the value (17.2) by $f(a)$. We apply this result to the function $F(z) = [f(z) - f(a)]/(z - a)$. It is not defined for $z = a$, but it satisfies the condition $\#\lim_{z \to a} f(z) = 0$. The #-limit of $F(z)$ as $z \to a$ is $f^{\#}(a)$. Hence there exists an #-analytic function which is equal to $F(z)$ for $z = a$ and equal to $f^{\#}(a)$ for $z = a$. Let us denote this function by $f_1(z)$. Repeating the process we can define an #-analytic function $f_2(z)$ which equals $[f_1(z) - f_2(a)]/(z - a)$ for $z \neq a$ and $f_1^{\#}(a)$ for $z = a$, and so on. The recursive scheme by which $f_n(z)$ is defined reads

$$\begin{align*} f(z) &= f(a) + (z-a)f_1(z) \\ f_1(z) &= f_1(a) + (z-a)f_2(z) \\ \multicolumn{2}{c}{\dots\dots\dots\dots\dots\dots\dots} \\ f_{n-1}(z) &= f_{n-1}(a) + (z-a)f_n(z). \end{align*} \qquad (17.3)$$

From these equations which are trivially valid also for $z = a$ we obtain

$$f(z) = f(a) + \mathrm{Ext} - \sum_{i=1}^{n} (z-a)^i f_i(z), n \in \mathbb{N}^{\#}. \qquad (17.4)$$

Differentiating $n$ times and setting $z = a$ we get

$$f^{(n)^\#}(a) = n! f_n(a). \qquad (17.5)$$

This determines the coefficients $f_n(a)$, and we obtain the following form of Taylor's theorem.

**Theorem 17.2.** If $f(z)$ is #-analytic in a region $\Delta$, containing $a$, then

$$f(z) = f(a) + \mathrm{Ext} - \sum_{i=1}^{n-1} \frac{f^{(i)^\#}(a)}{i!} (z-a)^i + f_n(z)(z-a)^n, \qquad (17.6)$$

where $n \in \mathbb{N}^\#$ and $f_n(z)$ is #-analytic in $\Omega$.

## 17.2. Zeros and Poles of #-Analytical Functions.
### 17.3. The Generalized Maximum Principle.

**Theorem 17.** (Generalized maximum principle.) If $f(z)$ is #-analytic and nonconstant in a region $\Omega$, then its absolute value $|f(z)|$ has no maximum in $\Omega$.

**Proof.** If $w_0 = f(z_0)$ is any value taken in $\Omega$, there exists a neighborhood $|w - w_0| < \varepsilon$ contained in the image of $\Omega$. In this #-neighborhood there are points of modulus $> w_0$, and hence $|f(z_0)|$ is not the maximum of $|f(z)|$.

**Theorem 17.** If $f(z)$ is defined and #-continuous on a #-closed bounded set $E$ and #-analytic on the interior of $E$, then the maximum of $|f(z)|$ on $E$ is assumed on the boundary of $E$.

**Proof.** Since $E$ is #-compact, $|f(z)|$ has a maximum on $E$. Suppose that it is assumed at $z_0$. If $z_0$ is on the boundary, there is nothing to prove. If $z_0$ is an interior point, then $|f(z_0)|$ is also the maximum of $|f(z)|$ in a disk $|z - z_0| < \delta$ contained in $E$. But this is not possible unless $f(z)$ is constant in the component of the interior of $E$ which contains $z_0$. It follows by #-continuity that $|f(z)|$ is equal to its maximum on the whole #-boundary of that component. This boundary is not empty and it is contained in the #-boundary of $E$. Thus the maximum is always assumed at a #-boundary point.

**Remark 17.1.** The generalized maximum principle can also be proved analytically, as a consequence of generalized Cauchy's integral formula. If the formula (22) is specialized to the case where $\gamma$ is a circle of center $z_0$ and radius $r$, we can write
$$ \zeta = z_0 + r(\mathrm{Ext}\,e^{i\theta}), \quad d^\#\zeta = ir(\mathrm{Ext}\,e^{i\theta}) \quad \text{and obtain for } z = z_0 $$

$$ f(z_0) = \frac{1}{2\pi i} \int_0^{2\pi} f(z_0 + r(\mathrm{Ext}\,e^{i\theta})) d^\#\theta. \quad (17.) $$

The formula (17.) shows that the value of an #-analytic function at the center of a circle is equal to the arithmetic mean of its values on the circle, subject to the condition

that the #-closed disk $|z - z_0| \le r$ is contained in the region of #-analyticity. From (34) we get the inequality

$$ |f(z_0)| \le \frac{1}{2\pi i} \int_0^{2\pi} |f(z_0 + r(\mathrm{Ext}\,e^{i\theta}))| d^\#\theta. \quad (17.) $$

Suppose that $|f(z_0)|$ were a maximum. Then we would have $|f(z_0 + r(\mathrm{Ext}\,e^{i\theta}))| \le |f(z_0)|$, and if the strict inequality held for a single value of $\theta$ it would hold, by #-continuity, on a whole arc. But then the mean value of $|f(z_0 + r(\mathrm{Ext}\,e^{i\theta}))|$ would be strictly less than $|f(z_0)|$, and (35) would lead to the contradiction $|f(z_0)| < |f(z_0)|$. Thus $|f(z)|$ must be constantly equal to $|f(z_0)|$ on all sufficiently small circles $|z - z_0| = r$ and, hence, in a neighborhood of $z_0$. It follows easily that $f(z)$ must reduce to a constant.
This reasoning provides a second proof of the generalized maximum principle. Consider now the case of a function $f(z)$ which is #-analytic in the #-open disk $|z| < R$ and #-continuous on the #-closed disk $|z| \le R$. If it is known that $|f(z)| \le M$ on $|z| = R$, then $|f(z)| \le M$ in the whole disk. The equality can hold only if $f(z)$ is a constant of absolute value $M$. Therefore, if it is known that $f(z)$ takes some value of modulus $< M$, it may be expected that a better estimate can be given.

**Theorem 17.** If $f(z)$ is #-analytic for $|z| < 1$ and satisfies the conditions $|f(z)| \le 1$, $f(0) = 0$, then $|f(z)| \le |z|$ and $|f^{#}(0)| \le 1$. If $|f(z)| = |z|$ for some $z \ne 0$, or if $|f^{#}(0)| = 1$, then $f(z) = cz$ with a constant $c \in \mathbb{C}_c^\#$ of absolute value 1.

**Proof.**

# Part II. $\mathbb{R}_c^\#$-Valued Lebesgue Integral

## 1. External $\mathbb{R}_c^\#$-Valued Lebesgue Measure

Let us consider a bounded interval $I \subset \mathbb{R}_c^\#$ with endpoints $a$ and $b$ ($a < b$). The length of this bounded interval $I$ is defined by $l(I) = b - a$. In contrast, the length of an unbounded interval, such as $(a, \infty^\#)$, $(-\infty^\#$, $b$) or $(-\infty^\#$, $\infty)$, is defined to be hyperinfinite. Obviously, the length of a line segment is easy to quantify. However, what should we do if we want to measure an arbitrary subset of $\mathbb{R}_c^\#$? Given a set $E \subset \mathbb{R}_c^\#$ of hyperreal numbers, we denote the Lebesgue measure of set $E$ by $\mu(E)$. To correspond with the length of a line segment, the measure of a set $A \subset \mathbb{R}_c^\#$ should keep the following properties:

(1) If $A$ is an interval, then $\mu(A) = l(A)$.

(2) If $A \subset B$, then $\mu(A) \le \mu(B)$.

(3) Given $A \subset \mathbb{R}_c^\#$ and $x_0 \in \mathbb{R}_c^\#$, define $A + x_0 = \{x + x_0 : x \in A\}$. Then $\mu(A) = \mu(A + x_0)$.

(4) If $A$ and $B$ are disjoint sets, then $\mu(A \cup B) = \mu(A) + \mu(B)$. If $\{A_i\}_{i \in \mathbb{N}^\#}$ is a hyperinfinite sequence of disjoint sets, then $\mu(\bigcup_{i \in \mathbb{N}^\#} A_i) = \sum_{i=1}^{\infty^\#} \mu(A_i)$.

## 1. External $\mathbb{R}_c^\#$-Valued Lebesgue outer measure

**Definition 1.1.** Let $E$ be a subset of $\mathbb{R}_c^\#$. Let $\{I_k\} \triangleq \{I_k\}_{k \in \mathbb{N}^\#}$ be a hyperinfinite sequence

of open intervals such that $E \subset \bigcup_{k \in \mathbb{N}^\#} A_k$ and let $\Sigma$ be a set of the all such hyperinfinite sequences. The external Lebesgue outer measure of $E$ is defined by

$$ \mu^*(E) = \inf_{\langle I_k \rangle \in \Sigma} \left\{ \sum_{k=1}^{\infty^\#} l(I_k) \right\}. \qquad (1.1) $$

Note that $0 \le \mu^*(E) \le \infty^\#$.

**Definition 1.2.** A set $E$ is #-countable if there exists an injective function $f$ from $E$ to the

gypernatural numbers $\mathbb{N}^\#$. If such an $f$ can be found that is also surjective (and therefore

bijective), then $E$ is called #-countably infinite or gyperinfinite, i.e. a set is #-countably
infinite if it has one-to-one correspondence with the set $\mathbb{N}^#$.

**Theorem 1.1.** The external Lebesgue outer measure has the following properties:

(a) If $E_1 \subseteq E_2$, then $\mu^*(E_1) \le \mu^*(E_2)$.

(b) The external Lebesgue outer measure of any #-countable set is zero.

(c) The external Lebesgue outer measure of the empty set is zero.

(d) The external Lebesgue outer measure is invariant under translation, that is,
$\mu^*(E + x_0) = \mu^*(E)$.

(e) Lebesgue outer measure is #-countably sub-additive, that is,

$$ \mu^*\left(\bigcup_{i=1}^{\infty} E_i\right) \le \sum_{i=1}^{\infty} \mu^*(E_i). \quad (14.2) $$

(f) For any interval $I$, $\mu^*(I) = l(I)$.

**Proof.** Part (a) is trivial.

For part (b) and (c), let $E = \{x_k | k \in \mathbb{Z}_+^#\}$ be a #-countably hyper infinite set.

Let $\varepsilon > 0, \varepsilon \approx 0$ and let $\varepsilon_k$ be a hyper infinite sequence of positive numbers such that

$\sum_{k=1}^{\infty} \varepsilon_k = \varepsilon/2$. Since $E \subseteq \bigcup_{k=1}^{\infty} (x_k - \varepsilon_k, x_k + \varepsilon_k)$, it follows that $\mu^*(E) \le \varepsilon$. Hence, $\mu^*(E) = 0$. Since $\emptyset \subseteq E$, then $\mu^*(\emptyset) = 0$.

For part (d), since each cover of $E$ by open intervals can generate a cover of $E + x_0$ by open intervals with the same length, then $\mu^*(E + x_0) \le \mu^*(E)$. Similarly, $\mu^*(E + x_0) \ge \mu^*(E)$, since $E + x_0$ is a translation of $E$ Therefore, $\mu^*(E + x_0) = \mu^*(E)$.

For part (e), if $\sum_{i=1}^{\infty} \mu^*(E_i) = \infty^#$, then the statement is trivial. Suppose that the sum is hyperfinite and let $\varepsilon > 0, \varepsilon \approx 0$. For each $i \in \mathbb{N}^#$, there exists a hyperinfinite sequence $\{I_i^k\}$ of open intervals such that $E_i \subseteq \bigcup_{k=1}^{\infty} I_i^k$ and $\sum_{k=1}^{\infty} l(I_i^k) < \mu^*(E_i) + \varepsilon/2^i$. Now $\{I_i^k\}$ is a double-indexed sequence of open intervals such that

$$ \bigcup_{i=1}^{\infty} E_i \subseteq \bigcup_{i=1}^{\infty} \bigcup_{k=1}^{\infty} I_i^k \text{ and} $$

$$ \sum_{i=1}^{\infty} \sum_{k=1}^{\infty} l(I_i^k) \le \sum_{i=1}^{\infty} (\mu^*(E_i) + \varepsilon/2^i) = \sum_{i=1}^{\infty} \mu^*(E_i) + \varepsilon. $$

Therefore, $\mu^*\left(\bigcup_{i=1}^{\infty} E_i\right) \le \sum_{i=1}^{\infty} \mu^*(E_i) + \varepsilon$. The result follows since $\varepsilon > 0, \varepsilon \approx 0$ was arbitrary.

For part (f), we need to prove $\mu^*(I) \le l(I)$ and $\mu^*(I) \ge l(I)$ respectively.

We can assume that $I = [a, b]$ where $a, b \in \mathbb{R}_c^#$.

First, we want to prove $\mu^*(I) \le l(I)$. Let $\varepsilon > 0, \varepsilon \approx 0$, we have

$$ I \subseteq (a, b) \cup (a - \varepsilon, a + \varepsilon) \cup (b - \varepsilon, b + \varepsilon). $$

$$ \text{Thus, } \mu^*(I) \le l(a, b) + l(a - \varepsilon, a + \varepsilon) + l(b - \varepsilon, b + \varepsilon) = (b - a) + 2\varepsilon + 2\varepsilon = b - a + 4\varepsilon. $$

As $\varepsilon > 0, \varepsilon \approx 0$ is arbitrary, we conclude that $\mu^*(I) \le b - a = l(I)$.

Then, we want to prove that $\mu^*(I) \ge l(I)$. Let $\{I_k\}$ be any sequence of open intervals that covers $I$. Since $I$ is compact, by the generalized Heine-Borel theorem, there is a gyperfinite subcollection $\{J_i\}$ with $1 \le i \le n$, $n \in \mathbb{N}^#$ of $I_k$ that still covers $I$. By reordering and deleting if necessary, we can assume that $a \in J_1 = (a_1, b_1), b_1 \in J_2 = (a_2, b_2), \dots, b_{n-1} \in J_n = (a_n, b_n)$, where $b_{n-1} \le b < b_n$.

We then can compute that

$$ b - a < b_n - a_1 = \mathrm{Ext} - \sum_{i=2}^{n} (b_i - b_{i-1}) + (b_1 - a_1) < \mathrm{Ext} - \sum_{i=1}^{n} l(J_i) \le \mathrm{Ext} - \sum_{i=1}^{\infty} l(I_k). $$

Therefore, $l(I) \le \mu^*(I)$. We can now conclude that $\mu^*(I) = l(I)$. This proves the result for closed and bounded intervals.
Suppose that $I = (a,b)$ is an open and bounded interval. Then, $\mu^*(I) \le l(I)$ as above and $b-a = \mu^*([a,b]) \le \mu^*((a,b)) + \mu^*(a) + \mu^*(b) = \mu^*((a,b))$.

Hence $l(I) \le \mu^*(I)$. The proof for half-open intervals is similar.

Finally, suppose that $I$ is a hyper infinite interval and let $M > 0$. There exists a bounded interval $J \subseteq I$ such that $\mu^*(J) = l(J) = M$ and it follows that $\mu^*(I) \ge \mu^*(J) = M$. Since $M > 0$ was arbitrary, $\mu^*(I) = \infty^\# = l(I)$.

This completes the proof.

## 14.2. External Lebesgue inner measure

In previous subsection, we have discussed external Lebesgue outer measure. There is

another external measure named external Lebesgue inner measure. Let's define the external inner measure and see some basic properties.

**Definition 14.2.** Let $E$ be a subset of $\mathbb{R}^\#$. The external inner measure of $E$ is defined by

$$ \mu^*(E) = \sup\{\mu(K) : K \subseteq E \text{ and } K \text{ is #-closed}\} \quad (14.3) $$

iff supremum in RHS of the (14.3) exists.

Recall that external Lebesgue outer measure of a set $E$ uses an infimum of the union of a sequence open sets that cover the set $E$, while external Lebesgue inner measure of a set $E$ uses a supremum of a set inside the set $E$. Then, it is obvious that

$$ \mu_*(E) \le \mu^*(E) \quad (14.4) $$

for any set $E$. Also, for $A \subseteq B, \mu_*(A) \le \mu^*(B)$.

**Theorem 14.2.** Let $A$ and $E$ be subsets of $\mathbb{R}_c^\#$.

(i) Suppose that $\mu^*(E) < \infty^\#$. Then $E$ is measurable if and only if $\mu_*(E) = \mu^*(E)$.

(ii) If $E$ is measurable and $A \subseteq E$, then $\mu(E) = \mu_*(A) + \mu^*(E \setminus A)$.

*Proof.* For part (i), suppose that $E$ is a measurable set and let $\varepsilon > 0, \varepsilon \approx 0$. According to Theorem 2.9, there exists a #-closed set $K$ such that $K \subseteq E$ and $\mu(E \setminus K) < \varepsilon$.

Thus, $\mu^*(E) \ge \mu_*(E) \ge \mu(K) > \mu(E) - \varepsilon = \mu^*(E) - \varepsilon$, which implies that the external inner

measure and external outer measure of $E$ are equal. Now let's prove the reverse direction. Suppose that $\mu_*(E) = \mu^*(E)$. Let $\varepsilon > 0, \varepsilon \approx 0$. Then

there exists a #-closed set $K$ and an #-open set $G$ such that $K \subseteq E \subseteq G$ and $\mu(K) > \mu_*(E) - \varepsilon/2$ and $\mu(G) < \mu^*(E) + \varepsilon/2$. Then we find that

$$ \mu^*(G \setminus E) \le \mu^*(G \setminus K) = \mu(G \setminus K) = \mu(G) - \mu(K) < \varepsilon. $$

According to Theorem 2.9, the set $E$ is measurable.

For part (2), let $\varepsilon > 0, \varepsilon \approx 0$. There exists a #-closed set $K \subseteq A$ such that

$$ \mu(K) > \mu_*(A) - \varepsilon. $$

Then, $\mu(E) = \mu(K) + \mu(E \setminus K) > \mu_*(A) - \varepsilon + \mu^*(E \setminus A)$ and it follows that $\mu(E) \ge \mu_*(A) + \mu^*(E \setminus A)$. According to Theorem 2.9, there exists a measurable set $B$ such that $E \setminus A \subseteq B \subseteq E$ and $\mu(B) = \mu^*(E \setminus A)$.

Since $E \setminus B \subseteq A$, it follows that $\mu_*(E \setminus B) \ge \mu_*(A)$. Thus,

$$ \mu(E) = \mu(B) + \mu(E \setminus B) = \mu^*(E \setminus A) + \mu(E \setminus B) \le \mu^*(E \setminus A) + \mu_*(A). $$

By combining these two inequalities, we can obtain $\mu(E) = \mu_*(A) + \mu^*(E \setminus A)$.

## 14.3. External Lebesgue measure

**Definition 14.3.** A set $E \subseteq \mathbb{R}_c^\#$ is Lebesgue measurable if for each set $A \subseteq \mathbb{R}_c^\#$, the equality $\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c)$ is satisfied. If $E$ is a Lebesgue
measurable set, then the external Lebesgue measure of E is its external Lebesgue outer measure and will be written as $\mu(E)$.

Since the external Lebesgue outer measure satisfies the property of subadditivity, then we always have $\mu^*(A) \le \mu^*(A \cap E) + \mu^*(A \cap E^c)$, $E^c = \mathbb{R}_c^\# \setminus E$ and we only need to check the reverse inequality.

Note that there is always a set E that can divide A into two mutually exclusive sets, $A \cap E$

and $A \cap E^c$. But only when $\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c)$ holds, the set E is Lebesgue

measurable. The latter theorem will show some properties of measurable sets.

**Theorem 14.3.** The collection of measurable sets defined on $\mathbb{R}_c^\#$ has the following properties:

(a) Both $\emptyset$ and $\mathbb{R}_c^\#$ are measurable.

(b) If E is measurable, then $E^c$ is measurable, where $E^c = \mathbb{R}_c^\# \setminus E$.

(c) If $\mu^*(E) = 0$, then E is measurable.

(d) If $E_1$ and $E_2$ are measurable, then $E_1 \cup E_2$ and $E_2 \cap E_2$ are measurable.

(e) If E is measurable, then $E + x_0$ is measurable.

**Proof.** For part (a), let $A \subseteq \mathbb{R}_c^\#$. Then

$$\mu^*(A \cap \emptyset) + \mu^*(A \cap E^c) = \mu^*(\emptyset) + \mu^*(A) = 0 + \mu^*(A) = \mu^*(A),$$

$$\mu^*(A \cap \mathbb{R}_c^\#) + \mu^*(A \cap E^c) = \mu^*(A) + \mu^*(\emptyset) = \mu^*(A) + 0 = \mu^*(A).$$

For part (b), if E is measurable, then for every set $A \subseteq \mathbb{R}_c^\#$, such that

$$\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c). \text{ Then,}$$

$$\mu^*(A \cap E^c) + \mu^*(A \cap (E^c) = \mu^*(A \cap E^c) + \mu^*(A \cap E) = \mu^*(A).$$

For part (c), let $A \subseteq \mathbb{R}_c^\#$. Since $\mu^*(E) = 0$ and $A \cap E \subseteq E$, then $\mu^*(A \cap E) = 0$.

We can obtain that $\mu^*(A) \ge \mu^*(A \cap E) = \mu^*(A \cap E) + \mu^*(A \cap E)$, which implies that $\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E)$ by Theorem 14.1 part (e).

For part (d), let $A \subseteq \mathbb{R}_c^\#$. Note that

$$A \cap (E_1 \cup E_2) = (A \cap E_1) \cup (A \cap E_2) = (A \cap E_1) \cup (A \cap E_1 \cap E_2)$$

Then, by De Morgan Law and Theorem 14.1 part (e), we know that

$$\begin{aligned} \mu^*(A) &= \mu^*(A \cap E_1) + \mu^*(A \cap E_1) = \\ &= \mu^*(A \cap E_1) + \mu^*(A \cap E_1 \cap E_2) + \mu^*(A \cap E_1 \cap E_2) \ge \mu^*(A \cap (E_1 \cup E_2)) + \\ &\quad + \mu^*(A \cap (E_1 \cup E_2)), \end{aligned}$$

showing that $E_1 \cup E_2$ is measurable. Since $E_1 \cap E_2 = (E_1 \cup E_2)$, then the set $E_1 \cap E_2$ is measurable by Theorem 14.1 part (b).

For part (e), let $A \subseteq \mathbb{R}_c^\#$. Then,

$$\begin{aligned} \mu^*(A) &= \mu^*(A - x_0) = \mu^*((A - x_0) \cap E) + \mu^*((A - x_0) \cap E^c) = \\ &\quad \mu^*(((A - x_0) \cap E) + x_0) + \mu^*(((A - x_0) \cap E^c) + x_0) = \\ &\quad \mu^*(A \cap (E + x_0)) + \mu^*(A \cap (E^c + x_0)). \end{aligned}$$

Therefore, $E + x_0$ is measurable.

**Lemma 14.1.** Let $E_i : 1 \le i \le n \in \mathbb{N}^\#$ be a gyperfinite collection of disjoint measurable sets. If $A \subseteq \mathbb{R}_c^\#$, then

$$\mu^*\left(\bigcup_{i=1}^{n} (A \cap E_i)\right) = \mu^*\left(A \cap \left(\bigcup_{i=1}^{n} E_i\right)\right) = \text{Ext}-\sum_{i=1}^{n} \mu^*(A \cap E_i).$$

**Proof.** We will prove this by the principle of mathematical induction. When $n = 1$, the equality holds. Suppose that the statement is valid for $n - 1$ disjoint measurable sets when $n > 1$. Then, when there are $n$ disjoint measurable sets,
$$
\begin{align*}
\mu^*(A \cap (\bigcup_{i=1}^n E_i)) &= \mu^*(A \cap (\bigcup_{i=1}^n E_i) \cap E_n) + \mu^*(A \cap (\bigcup_{i=1}^n E_i) \cap E_n^c) \\
&= \mu^*(A \cap E_n) + \mu^*(A \cap (\bigcup_{i=1}^{n-1} E_i)) \\
&= \mu^*(A \cap E_n) + \mathrm{Ext}\sum_{i=1}^{n-1} \mu^*(A \cap E_i) = \mathrm{Ext}\sum_{i=1}^n \mu^*(A \cap E_i).
\end{align*}
$$

Note that when $A = \mathbb{R}_c^\#$, $\mu(\bigcup_{i=1}^n E_i) = \mathrm{Ext}\sum_{i=1}^n \mu(E_i)$.

**Theorem 14.4.** If {$E_i$}$_{i \in \mathbb{N}^\#}$ is a hyper infinite sequence of disjoint measurable sets, then

$$
\mu\left(\bigcup_{i=1}^{\infty} E_i\right) = \mathrm{Ext}\sum_{i=1}^{\infty} \mu(E_i). \qquad (14.5)
$$

**Proof.** According to Lemma 14.1, $\mathrm{Ext}\sum_{i=1}^n \mu(E_i) = \mu\left(\bigcup_{i=1}^n E_i\right) \le \mu\left(\bigcup_{i=1}^{\infty} E_i\right)$ for each positive integer $n \in \mathbb{N}^\#$, which implies that $\mathrm{Ext}\sum_{i=1}^\infty \mu(E_i) \le \mu\left(\bigcup_{i=1}^\infty E_i\right)$.

By #-countably subadditive property, $\mathrm{Ext}\sum_{i=1}^{\infty} \mu(E_i) \ge \mu\left(\bigcup_{i=1}^{\infty} E_i\right)$.

$$
\text{Therefore, } \mathrm{Ext}\sum_{i=1}^{\infty} \mu(E_i) = \mu\left(\bigcup_{i=1}^{\infty} E_i\right).
$$

The previous theorem shows that if A and B are disjoint measurable sets, then $\mu(A \cup B) = \mu(A) + \mu(B)$. If {$A_i$}$_{i \in \mathbb{N}^\#}$ is a hyper infinite sequence of disjoint measurable sets, then $\mu(\bigcup_{i=1}^\infty E_i) = \mathrm{Ext}\sum_{i=1}^\infty \mu(A_i)$. As so far, we have already seen that when the sets are measurable, Lebesgue measure satisfies property (1),(2),(3) and (4). But what kinds of sets are measurable? Certainly every interval is measurable.

**Theorem 14.5.** Every interval $[a, b] \subset \mathbb{R}^\#$ is measurable.

**Theorem 14.6.** If {$E_i$}$_{i \in \mathbb{N}^\#}$ is a hyper infinite sequence of measurable sets, then

$\infty$
S
$i=1$
E i and

$\infty$
T
$i=1$
E i
are measurable sets.

**Definition 14.4.** Let $f$ be a function from $E \subset \mathbb{R}_c^\#$ into $\mathbb{R}_c^\# \cup (-\infty^\#, \infty^\#)$. The function $f$ is (Lebesgue) measurable if
## 15. External Lebesgue Integral

Let $(\mathbb{R}, B, \mu)$ be the standard Lebesgue space on $\mathbb{R}$. Our internal starting point could be the internal measure space $(\ast\mathbb{R}, \ast B, \ast\mu)$. By transfer we can write down internal Lebesgue integrals

$$ \mathcal{L}[\ast f(t)] = \int_A \ast f(t) d^* \mu(t), $$

where $A \in \ast B$ and $f: \mathbb{R} \to \mathbb{R}$.

### 15.1. Lebesgue Integral of a $\mathbb{R}^c$-valued external function $f(x)$.

First, in particular, we need external function that can help us distinguish whether a given value $x$ is in the measurable set $A_i$. We call this function the characteristic function. The following statement is the formal definition of characteristic function and introduces the simple function.

**Definition 15.1.** For any set $A$, the function

$$ \chi_A(x) = \begin{cases} 1, & x \in A, \\ 0, & \text{otherwise} \end{cases} \qquad (15.1) $$

is called the characteristic function of set $A$. A linear combination of characteristic functions,

$$ \phi(x) = \sum_{i=1}^{n} a_i \chi_{A_i}(x) \qquad (15.2) $$

is called a simple function if the sets $A_i$ are measurable.

For a function $f: \mathbb{R}^c \to \mathbb{R}^c$ defined on a measurable set $A$ that takes no more than $\text{gyper finitely many distinct values } a_1, \dots, a_n, n \in \mathbb{N}^c$ the function $f$ can always be written as a simple function

$$ f(x) = \sum_{i=1}^{n} a_i \chi_{A_i}(x), \qquad (15.3) $$

where $A_i = \{x \in A | f(x) = a_i\}$. That is a simple function of the first kind.

Therefore, simple functions can be thought of as dividing the range of $f$, where resulting sets $A_i$ may or may not be intervals.

Let us pause for a second. We want to ask ourselves: is the simple function $\phi(x)$ unique? The answer is no. Because we might define different disjoint sets that have a same function value. The simplest expression is

$$ \phi(x) = \sum_{i=1}^{n} a_i \chi_{A_i}(x) \qquad (15.4) $$

where $A_i = \{x \in A | \phi(x) = a_i\}$. At this case, the constants $a_i$ are distinct, the sets $A_i$ are disjoint and we call that representation the canonical representation of $\phi$.

Then, for simple functions, we define the Lebesgue integral as follows:

**Definition 15.2.** If $\phi(x) = \sum_{i=1}^{n} a_i \chi_{A_i}(x)$ is a simple function and $\mu(A_i)$ is gyperfinite for all $i$, then the Lebesgue integral of $\phi(x)$ is defined as

$$ \int_E \phi(x) = \sum_{i=1}^{n} a_i \chi_{A_i}(x). \qquad (15.5) $$
**Definition 15.3.** Suppose $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is a bounded function defined on a measurable set $E$ with giperfinite measure. We define the upper and lower Lebesgue integrals if exist, respectively, as

$$I_L^\#(f) = \int_E \inf\{\phi(x)|\phi \text{ is simple and } \phi \ge f\} \quad (15.6)$$

and

$$I_{\#L}(f) = \int_E \sup\{\phi(x)|\phi \text{ is simple and } \phi \le f\}. \quad (15.7)$$

If (i) the quantity $I_L^\#(f)$ and $I_{\#L}(f)$ exist and (ii) $I_L^\#(f) = I_{\#L}(f)$, then the function $f$ is called Lebesgue integrable over set $E$ and the external Lebesgue integral of $f$ over set $E$ is denoted by $I_L(f) = \int_E fdx$.

**The Lebesgue Integral for Simple Functions of the second kind**

Let $\varphi(x)$ be some simple external function of the second kind which takes on the gyperinfinitely many distinct values $y_1, \dots, y_n, \dots, n \in \mathbb{N}^\#$, $y_i \neq y_j$ for $i \neq j$.

It is natural to define the integral of the function $\varphi(x)$ over the set $E$ by the equation

$$\int_E \varphi(x) d^\# \mu = \sum_{n \in \mathbb{N}^\#} y_n \mu\{x | x \in E, \varphi(x) = y_n\}. \quad (15.8)$$

**Definition 15.4.** The simple function $\varphi(x)$ of the second kind is called integrable (with respect to the measure $\mu$) over the set $E$ if the gyperinfinite series (15.8) #-converges absolutely.

If $\varphi(x)$ is #-integrable, then the sum of the series (15.8) is called the integral of $\varphi(x)$ over the set $E$.

**Remark 15.1.** Note that in definition 15.4 we assume that all the $y_n$ are different. One can, however, represent the value of the integral of a simple function as a sum of products of the form $c_k\mu(B_k)$ and not assume that all the $c_k$ are different.

**Lemma 15.1.** Let $A = \cup_k B_k, B_i \cap B_j = \emptyset$ for $i \neq j$, and assume that on each set $B_k$ the function $f(x)$ takes on only one value $c_k$. Then

$$\int_A \varphi(x) d^\# \mu = \sum_{k \in \mathbb{N}^\#} c_k \mu(B_k). \quad (15.9)$$

moreover, the function $f(x)$ is integrable over $A$ if and only if the gyper infinite series (15.9) #-converges absolutely.

**Proof.** It is easy to see that every set $A_n = \{x | x \in A, f(x) = y_n\}$ is the union of those $B_k$ for which $c_k = y_n$. Therefore

$$\sum_{n \in \mathbb{N}^\#} y_n \mu(A_n) = \sum_{n \in \mathbb{N}^\#} y_n \sum_{c_k=y_k} \mu(B_k) = \sum_{k \in \mathbb{N}^\#} c_k \mu(B_k). \text{ Since the measure is non-negative, } \sum_{n \in \mathbb{N}^\#} |y_n| \mu(A_n) = \sum_{n \in \mathbb{N}^\#} |y_n| \sum_{c_k=y_k} \mu(B_k) = \sum_{k \in \mathbb{N}^\#} |c_k| \mu(B_k).$$

i.e., the series $\sum_{n \in \mathbb{N}^\#} y_n \mu(A_n)$ and $\sum_{k \in \mathbb{N}^\#} |c_k| \mu(B_k)$ both either #-converge absolutely or #-diverge.

Let us consider some properties of the external Lebesgue integral for simple external functions:

$$\int_A f(x) d^\# \mu + \int_A g(x) d^\# \mu = \int_A [f(x) + g(x)] d^\# \mu \quad (15.10)$$

moreover, from the existence of the integrals on the left-hand side it follows that the integrals on the right-hand side exist.

To prove this assume that $f(x)$ takes on the values $f_i$, on the sets $F_i \subseteq A$, and $g(x)$ the values $g_i$, on the sets $G_i \subseteq A$, since

$$J_1 = \int_A f(x) d^\# \mu = \sum_{i \in \mathbb{N}^\#} f_i \mu(F_i) \quad (15.11)$$
and

$$J_2 = \int_A g(x) d^\# \mu = \sum_{i \in \mathbb{N}^\#} g_i \mu(G_j). \qquad (15.12)$$

Then, by the Lemma 14.1 we get

$$J = \int_A [f(x) + g(x)] d^\# \mu = \sum_{i \in \mathbb{N}^\#} \sum_{j \in \mathbb{N}^\#} [f_i + g_j] \mu(F_i \cap G_j), \qquad (15.13)$$

where

$$\mu(F_i) = \sum_{j \in \mathbb{N}^\#} \mu(F_i \cap G_j), \mu(G_j) = \sum_{i \in \mathbb{N}^\#} \mu(F_i \cap G_j). \qquad (15.14)$$

From the absolute #-convergence of the series (15.11)-(15.12) it follows the absolute #-convergence of the series (15.13); here $J = J_1 + J_2$.

For any constant $k \in \mathbb{R}_c^\#$

$$k \int_A f(x) d^\# \mu = \int_A [kf(x)] d^\# \mu \qquad (15.15)$$

moreover, the existence of the integral on the left-hand side implies the existence of the

integral on the right. A simple function $f(x)$ which is bounded on the set $A \subset \mathbb{R}_c^\#$ is #-integrable over A; moreover, if $|f(x)| \le M \in \mathbb{R}_c^\#$ on A, then

$$\left| \int_A f(x) d^\# \mu \right| \le M \mu(A). \qquad (15.16)$$

# 15. General Definition and Basic Properties of the external Lebesgue Integral.

**Definition.15.1.** We shall say that the function $f(z)$ is #-integrable over the set $A \subset \mathbb{R}_c^\#$, if

there exists a hyper infinite sequence of simple functions $f_n(z), n \in \mathbb{N}^\#$ which are #-integrable over A and #-converge uniformly to $f(x)$. We shall denote the #-limit

$$J = \#\text{-lim}_{n \to \infty} \int_A f_n(x) d^\# \mu \qquad (15.1)$$

by

$$\int_A f(x) d^\# \mu. \qquad (15.2)$$

and call it the integral of the external function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ over the set A.

This definition 15.1 is correct if the following conditions are satisfied:

1. The #-limit (15.1) for any uniformly #-convergent hyper infinite sequence of simple functions which are #-integrable over A exists.

2. This #-limit for a given function $f(x)$ does not depend on the choice of the hyper infinite

sequence $\{f_n(x)\}_{n \in \mathbb{N}^\#}$.

3. For simple functions the definitions of #-integrability and #-integral are equivalent to those given in section 14.

Notice that all these conditions are indeed satisfied.

To prove the first it suffices to note that by properties for #-integrals of simple functions,

$$\left| \int_A f_n(x) d^\# \mu - \int_A f_m(x) d^\# \mu \right| \le \mu(A) \sup_{x \in A} |f_n(x) - f_m(x)|. \qquad (15.3)$$
To prove the second condition, we must consider the two hyperinfinite sequences ${f_n(x)}_{n \in \mathbb{N}^\#}$ and ${f'_n(x)}_{n \in \mathbb{N}^\#}$, and use the inequality

$$ \left| \int_A f_n(x) d^\# \mu - \int_A f'_n(x) d^\# \mu \right| \le \mu(A) \left[ \sup_{x \in A} |f_n(x) - f(x)| + \sup_{x \in A} |f'_n(x) - f(x)| \right]. \quad (15.4) $$

Finally, to prove the third condition it suffices to consider the hyperinfinite sequence $f_n(x) = f(x)$.

**The basic properties of the external Lebesgue \#-integral.**

**Theorem 15.1.**

$$ \int_A 1 \cdot d^\# \mu = \mu(A). \quad (15.5) $$

**Proof.** Immediately from the definition of the \#-integral.

**Theorem 15.2.** For any constant $k \in \mathbb{R}_c^\#$

$$ k \int_A f(x) d^\# \mu = \int_A [kf(x)] d^\# \mu \quad (15.6) $$

where the existence of the \#-integral on the left-hand side implies the existence of the \#-integral on the right.

**Proof.** The proof is obtained from property (8.15) by proceeding to the \#-limit for an \#-integral of simple functions.

**Theorem 15.3.** Assume that $f(x)$ and $g(x)$ are \#-integrable over A then $f(x) + g(x)$ \#-integrable over A and

$$ \int_A f(x) d^\# \mu + \int_A g(x) d^\# \mu = \int_A [f(x) + g(x)] d^\# \mu \quad (15.7) $$

Let ${f_i(x)}_{i=1}^n, n \in \mathbb{N}^\#$ be a hyperfinite sequence such that any $f_i(x)$ is \#-integrable over A

then $\sum_{i=1}^n f_i(x)$ is \#-integrable over A and

$$ \sum_{i=1}^{n} \int_{A} f_{i}(x) d^{\#} \mu = \int_{A} \left[ \sum_{i=1}^{n} f_{i}(x) \right] d^{\#} \mu \quad (15.8) $$

where the existence of the \#-integrals on the left implies the existence of the \#-integral on the right.

**Proof.** The proof of (15.7) is obtained from property A) by proceeding to the \#-limit for an

\#-integral of simple functions.

**Theorem 15.4.** A function $f: A \to \mathbb{R}_c^\#$ which is hyperbounded on the set A is \#-integrable

over A.

**Proof.** The proof is obtained from property C) by proceeding to the limit for an integral

of simple functions.

**Theorem 15.5.** If $f(x) \ge 0$, then

$$ \int_A f(x) d^\# \mu \ge 0 \quad (15.9) $$

assuming that the \#-integral exists.

**Proof.** For simple functions this follows immediately from the definition; for the general case the proof is based on the possibility of approximating non-negative functions by
non-negative simple functions

**Corollary 15.1.** If $f(x) \ge g(x)$, then

$$ \int_A f(x) d^\# \mu \ge \int_A g(x) d^\# \mu. \qquad (15.10) $$

## References

[1] Foukzon, J. (2021). Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I). Journal of Advances in Mathematics and Computer Science, 36(2), 73-88. https://doi.org/10.9734/jamcs/2021/v36i230339

[2] Foukzon, J. (2021). Set Theory INC\_\# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions. Journal of Advances in Mathematics and Computer Science, 36(4), 90-112. https://doi.org/10.9734/jamcs/2021/v36i430359

[3] Foukzon, J. (2021). Set Theory INC\_\# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem. Journal of Advances in Mathematics and Computer Science, 36(8), 70-119. https://doi.org/10.9734/jamcs/2021/v36i830394

[4] A. Robinson, Non-standard analysis. (Reviwed re-edition of the 1st edition of (1966) Princeton: Princeton University Press, 1996.

[4] Stroyan, K.D., Luxemburg, W.A.J. Introduction to the theory of infinitesimals. New York: Academic Press (1st ed.), 1976

[5] S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Dover Books on Mathematics), February 26, 2009 Paperback: 526 pages ISBN-10: 0486468992, ISBN-13: 978-0486468990

[6] A. Lightstone, A. Robinson,

[7] Bas C. van Fraassen, Formal Semantics and Logic, Hardcover – January, 1971 256 pages. ISBN-10:0024226904 ISBN-13:978-0024226907

[8] E. Mendelson, Introduction to Mathematical Logic, Fourth Edition, CRC Press, Jun 1997 - Mathematics - 440 pages

[9] Foukzon, J., Relevant First-Order Logic LP# and Curry's Paradox Resolution, Pure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015, Pages: 6-12 DOI: 10.11648/j.pamj.s.2015040101.12 arXiv:0804.4818 [math.LO] https://arxiv.org/abs/0804.4818v2

[6] J. Foukzon, Non-archimedean analysis on the extended hyperreal line *R\_d and the solution of some very old transcendence conjectures over the field Q. arXiv:0907.0467 [math.GM]

[7] F. Wattenberg, [0, \infty] -valued, translation invariant measures on the Dedekind completion of *R. Pacific J. Math. Volume 90, Number 1 (1980), 223-247.

[8] H. Gonshor, Remarks on the Dedekind completion of a nonstandard model of the reals. Pacific J. Math. Volume 118, Number 1 (1985), 117-132.
# Appendix A. Bivalent Hyper Infinitary first-order logic $^{2}L_{\infty}^{#}$ with restricted rules of conclusion. Generalized Deduction Theorem.

Hyper infinitary language $L_{\infty}^{#}$ are defined according to the length of hyper infinitary
conjunctions/disjunctions as well as quantification it allows. In that way, assuming a
supply of $\kappa < \aleph_{0}^{#} = card(\mathbb{N}^{#})$ variables to be interpreted as ranging over a nonempty
domain, one includes in the inductive definition of formulas an infinitary clause for
conjunctions and disjunctions, namely, whenever the hypernaturals indexed hyper
infinite sequence $\{A_{\delta}\}_{\delta \in \mathbb{N}^{#}}$ of formulas has length less than $\kappa$, one can form the
hyperfinite conjunction/disjunction of them to produce a formula. Analogously, whenever
an hypernaturals indexed sequence of variables has length less than $\lambda$, one can
introduce one of the quantifiers $\forall$ or $\exists$ together with the sequence of variables in front of
a formula to produce a new formula. One also stipulates that the length of any
well-formed formula is less than $\aleph_{0}^{#}$ itself.

The syntax of bivalent hyper infinitary first-order logics $^{2}L_{\infty}^{#}$ consists of a (ordered) set of sorts and a set of function and relation symbols, these latter together with the corresponding type, which is a subset with less than $\aleph_{0}^{#} = card(\mathbb{N}^{#})$ many sorts. Therefore, we assume that our signature may contain relation and function symbols on $\gamma < \aleph_{0}^{#}$ many variables, and we suppose there is a supply of $\kappa < \aleph_{0}^{#}$ many fresh variables of each sort. Terms and atomic formulas are defined as usual, and general formulas are defined inductively according to the following rules.

If $\phi, \psi, \{\phi_{\alpha} : \alpha < \gamma\}$ (for each $\gamma < \kappa$) are formulas of $L_{\infty}^{#}$, the following are also formulas:

(i) $\bigwedge_{\alpha<\gamma} \phi_{\alpha}, \bigwedge_{\alpha\leq\gamma} \phi_{\alpha},$

(ii) $\bigvee_{\alpha<\gamma} \phi_{\alpha}, \bigvee_{\alpha\leq\gamma} \phi_{\alpha},$

(iii) $\phi \rightarrow \psi, \phi \wedge \psi, \phi \vee \psi, \neg \phi$

(iv) $\forall_{\alpha<\gamma} x_{\alpha}\phi$ (also written $\forall\mathbf{x}_{\gamma}\phi$ if $\mathbf{x}_{\gamma} = \{x_{\alpha} : \alpha < \gamma\}$),

(v) $\exists_{\alpha<\gamma} x_{\alpha}\phi$ (also written $\exists\mathbf{x}_{\gamma}\phi$ if $\mathbf{x}_{\gamma} = \{x_{\alpha} : \alpha < \gamma\}$),

(vi) the statement $\bigwedge_{\alpha<\gamma} \phi_{\alpha}$ holds if and only if for any $\alpha$ such that $\alpha < \gamma$

the statement holds $\phi_{\alpha}$,

(vii) the statement $\bigvee_{\alpha<\gamma} \phi_{\alpha}$ holds if and only if there exist $\alpha$ such that $\alpha < \gamma$

the statement holds $\phi_{\alpha}$.

**Definition 1.**[7]. A valuation of a syntactic system is a function that as signs $\vdash$ (true)
to some of its sentences, and/or $\perp$ (false) to some of its sentences. Precisely, a
valuation maps a nonempty subset of the set of sentences into the set $\{\vdash, \perp\}$.
We call a valuation bivalent iff it maps all the sentences into $\{\vdash, \perp\}$.

**Definition 2.**[7]. *L* is a bivalent propositional language iff its admissible valuations are the functions *v* such that for all sentences *A*, *B* of *L*,

(a) $v(A) \in \{\top, \bot\}$

(b) $v(\neg A) = \top \text{ iff } v(A) = \bot$

(c) $v(A \wedge B) = \top \text{ iff } v(A) = v(B) = \top.$

(d) by definition of the implication $A \Rightarrow B$ the following truth table holds
$v(A) \quad v(B) \quad v(A \Rightarrow B)$

(1) $\top \quad \top \quad \top$

(2) $\top \quad \bot \quad \bot$

(3) $\bot \quad \top \quad \top$

(4) $\bot \quad \bot \quad \top$

Truth table 1.

**Remark 1.** Note that in the case (4) on a truth table 1

In this case we call implication $A \Rightarrow B$ a weak implication and abbreviate

$$A \Rightarrow_B B \qquad (1)$$

We call a statement (1) as a weak statement and often abbreviate $v(A \Rightarrow B) = \top_w$ instead (1).

**Definition 3.** [7-8]. A is a valid (logically valid) sentence (in symbols, $\vdash A$) in $L$ iff every admissible valuation of $L$ satisfies A.

The axioms of hyper infinitary first-order logic $^2L_{\infty}^{\#}$ consist of the following schemata:

**I. Logical axiom**

A 1. $A \to [B \to A]$

A 2. $[A \to [B \to C]] \to [[A \to B] \to [A \to C]]]$

A 3. $[\neg B \to \neg A] \to [A \to B]$

A 4. $[\bigwedge_{i<\alpha} [A \to A_i]] \to [A \to \bigwedge_{i<\alpha} A_i], \alpha \in \mathbb{N}^\#$

A 5. $[\bigwedge_{i<\alpha} A_i] \to A_j, \alpha \in \mathbb{N}^\#$

A 6. $[\forall \mathbf{x}[A \to B] \to [A \to \forall \mathbf{x}B]]$ provided no variable in $\mathbf{x}$ occurs free in $A$;

A 7. $\forall \mathbf{x}A(\mathbf{x}) \to S_f(A)$, where $S_f(A)$ is a substitution based on a function $f$ from $\mathbf{x}$ to the terms of the language; in particular:

A 7'. $\forall x_i[A(x_i)] \Rightarrow A(\mathbf{t})$ is a wff of $^2L_{\infty}^{\#}$ and $\mathbf{t}$ is a term of $^2L_{\infty}^{\#}$ that is free for $x_i$ in $A(x_i)$. Note here that $\mathbf{t}$ may be identical with $x_i$; so that all wffs $\forall x_i A \Rightarrow A$ are axioms by virtue of axiom (7), see [8].

A 8. Gen (Generalization).

$\forall x_i B$ follows from $B$.

**II. Restricted rules of conclusion.**

Let $\mathcal{F}_{wff}$ be a set of the all closed wffs of $L_{\infty}^{\#}$.

**R1.** RMP (Restricted Modus Ponens).

There exist subsets $\Delta_1, \Delta_2 \subset \mathcal{F}_{wff}$ such that the following rules are satisfied.

From $A$ and $A \Rightarrow B$, conclude $B$ iff $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$, where $\Delta_1, \Delta_2 \subset \mathcal{F}_{wff}$.

In particular for any $A, B \in \mathcal{F}_{wff}: A \Rightarrow_B B \in \Delta_2$.

If $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$ we also abbreviate by $A, A \Rightarrow B \vdash_{RMP} B$.

**R2.** RMT (Restricted Modus Tollens)

There exist subsets $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{wff}$ such that the following rules are satisfied.

$P \Rightarrow Q, \neg Q \vdash_{RMT} \neg P$ iff $P \notin \Delta'_1$ and $(P \Rightarrow Q) \notin \Delta'_2$, where $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{wff}$.

In particular for any $P, Q \in \mathcal{F}_{wff}: P \Rightarrow_Q Q \in \Delta_2$.

**Remark 2.** Note that RMP and RMT easily prevent any paradoxes of naive Cantor set theory (NC), see [1],[9].

**III.** Additional derived rule of conclusion.
**Particularization rule (RPR)**

Remind that canonical unrestricted particularization rule (UPR) reads

**UPR:** If **t** is free for *x* in *B*(*x*), then ∀*x*[*B*(*x*)] ⊢ *B*(**t**), see [8].

**Proof.** From ∀x[B(x)] and the instance ∀x[B(x)] ⇒ B(t) of axiom (A7), we obtain B(t)
by unrestricted modus ponens rule. Since *x* is free for *x* in *B*(*x*), a special case of
unrestricted particularization rule is: ∀xB ⊢ B.

**Definition 4.** Any formal theory *L* with a hyper infinitary language *L*<sub>∞</sub><sup>#</sup> is defined when the following conditions are satisfied:

1. A hyper infinite set of symbols is given as the symbols of L. A finite or hyperfinite sequence of symbols of L is called an expression of L.

2. There is a subset of the set of expressions of *L* called the set of *well formed formulas (wffs) of* *L*. There is usually an effective procedure to determine whether a given expression is a wff.

3. There is a set of wfs called the set of axioms of L. Most often, one can effectively decide whether a given wff is an axiom; in such a case, L is called an axiomatic theory.

4. There is a *finite* set $R_1, \dots, R_n$, of relations among wffs, called rules of conclusion. For each $R_i$, there is a unique positive integer $j$ such that, for every set of $j$ wfs and each wff $B$, one can effectively decide whether the given $j$ wffs are in the relation $R_i$ to $B$, and, if so, $B$ is said to follow from or to be a direct consequence of the given wffs by virtue of $R_j$.

**Definition 5.** A proof in *L* is a finite or *hyperfinite* sequence *B*₁, ..., *B*ₖ, *k* ∈ N# of wffs such that for each *i*, either *B*ᵢ is an axiom of *L* or *B*ᵢ is a direct consequence of some of the preceding wffs in the sequence by virtue of one of the rules of inference of *L*.

**Definition 6.** A theorem of *L* is a wff *B* of *Y* such that *B* is the last wff of some proof in *L*. Such a proof is called a proof of *B* in *L*.

**Definition 7.** A wff *E* is said to be a consequence in *L* of a set of Γ of wffs if and only if there is a finite or *hyperfinite* sequence *B*₁, ..., *B*ₖ, *k* ∈ N# of wffs such that *E* is *B*ₖ and, for each *i*, either *B*ᵢ is an axiom or *B*ᵢ is in Γ, or *B*ᵢ is a direct consequence by some rule of inference of some of the preceding wffs in the sequence. Such a sequence is called a proof (or deduction) *E* from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ *E* as an abbreviation for *E* as a consequence of Γ.
In order to avoid confusion when dealing with more than one theory, we write
Γ ⊢<sub>*L*</sub> *E*, adding the subscript *L* to indicate the theory in question.
If Γ is a finite or *hyperfinite* set {*H*ᵢ}₁≤i≤m, *m* ∈ N# we write *H*₁, ..., *H*ₘ ⊢ *E* instead
of {*H*ᵢ}₁≤i≤m ⊢ *E*.

**Lemma 1.** [8]. ⊢ *B* ⇒ *B* for all wffs *B*.

**Theorem 1.** (Generalized Deduction Theorem 1). If Γ is a set of wffs and B and E are wffs, and Γ, B ⊢ E, then Γ ⊢ B ⇒<sub>s</sub> E. In particular, if B ⊢ E then ⊢ B ⇒ E.

**Proof.** Let $E_1, \dots, E_n, n \in \mathbb{N}^#$ be a proof of $E$ form $\Gamma \cup \{B\}$, where $E_n$ is $E$.
Let us prove, by *hyperfinite induction* on $j$, that $\Gamma \vdash B \Rightarrow_s E_j$ for $1 \le j \le n$.
First of all, $E_1$ must be either in $\Gamma$ or an axiom of $L$ or $B$ itself.

By axiom schema A1, $E_1 \Rightarrow_s (B \Rightarrow_s E_1)$ is an axiom. Hence, in the first two cases,
by MP, $\Gamma \vdash B \Rightarrow_s E_1$ For the third case, when $E_1$ is $B$, we have $\vdash B \Rightarrow_s E_1$ by Lemma 1, and, therefore, $\Gamma \vdash B \Rightarrow_s E_1$. This takes care of the case $j = 1$. Assume now that: $\vdash B \Rightarrow_s E_k$ for all $k < j, j \in \mathbb{N}^#$. Either $E_j$ is an axiom, or $E_j$ is in $\Gamma$, or $E_j$ is $B$, or $E_j$ follows by modus ponens from some $E_l$ and $E_m$ where $l < j, m < j$, and $E_m$ has the form $E_l \Rightarrow_s E_j$. In the first three cases, $\Gamma \vdash B \Rightarrow_s E_j$ as in the case $j = 1$ above. In the last case, we have, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s E_l$ and $\Gamma \vdash B \Rightarrow_s (E_l \Rightarrow_s E_j)$. But, by axiom schema (A2), $\vdash B \Rightarrow_s (E_l \Rightarrow_s E_j) \Rightarrow_s ((B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j))$

Hence, by MP, $\Gamma \vdash (B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j)$ and, again by MP, $\Gamma \vdash B \Rightarrow_s E_j$. Thus, the proof by hyperfinite induction is complete.

The case $j = n \in \mathbb{N}^#$ is the desired result. Notice that, given a deduction of $E$ from $\Gamma$ and $B$, the proof just given enables us to construct a deduction of $B \Rightarrow_s E$ from $\Gamma$. Also note that axiom schema A3 was not used in proving the generalized deduction theorem.

**Remark 3.** For the remainder of the chapter, unless something is said to the contrary, we shall omit the subscript $L$ in $\vdash_L$. In addition, we shall use $\Gamma, B \vdash E$ to stand for $\Gamma \cup \{B\} \vdash E$. In general, we let $\Gamma, B_1, \dots, B_n \vdash E$ stand for $\Gamma \cup \{B_i\}_{1 \le i \le n} \vdash E$.

**Remark 4.** We shall use the terminology proof, theorem, consequence, axiomatic, etc. and notation $\Gamma \vdash E$ introduced above.

**Proposition 1.** Every wff $B$ of $K$ that is an instance of a tautology is a theorem of $K$, and it may be proved using only axioms A1-A3 and MP.

**Proposition 2.** If $E$ does not depend upon $B$ in a deduction showing that $\Gamma, B \vdash E$, then $\Gamma \vdash E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$, in which $E$ does not depend upon $B$. In this deduction, $D_n$ is $E$. As an inductive hypothesis, let us assume that the proposition is true for all deductions of length less than $n \in \mathbb{N}^#$. If $E$ belongs to $\Gamma$ or is an axiom, then $\Gamma \vdash E$. If $E$ is a direct consequence of one or two preceding wffs by Gen or MP, then, since $E$ does not depend upon $B$, neither do these preceding wfs. By the inductive hypothesis, these preceding wfs are deducible from $\Gamma$ alone. Consequently, so is $E$.

**Theorem 2.** (Generalized Deduction Theorem 2). Assume that, in some deduction showing that $\Gamma, B \vdash E$, no application of Gen to a wff that depends upon $B$ has as its quantified variable a free variable of $B$. Then $\Gamma \vdash B \Rightarrow_s E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$ satisfying the assumption of this theorem. In this deduction, $D_n$ is $E$. Let us show by hyperfinite induction that $\Gamma \vdash B \Rightarrow_s D_i$ for each $i \le n \in \mathbb{N}^#$. If $D_i$ is an axiom or belongs to $\Gamma$, then $\Gamma \vdash B \Rightarrow_s D_i$, since $D_i \Rightarrow_s (B \Rightarrow_s D_i)$ is an axiom. If $D_i$ is $B$, then $\Gamma \vdash B \Rightarrow_s D_i$, since, by Proposition 1, $\vdash B \Rightarrow_s B$. If there exist $j$ and $k$ less than $i$ such that $D_k$ is $\vdash D_j \Rightarrow_s D_i$, then, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$ and $\Gamma \vdash B \Rightarrow_s (D_j \Rightarrow_s D_i)$. Now, by axiom A2,

$\vdash B \Rightarrow_s (D_j \Rightarrow_s D_i) \Rightarrow_s ((B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s D_i))$. Hence, by MP twice, $\Gamma \vdash B \Rightarrow_s D_i$. Finally, suppose that there is some $j < i$ such that $D_i$ is $\forall x_k D_j$. By the inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$, and, by the hypothesis of the theorem, either $D_j$ does not depend upon $B$ or $x_k$ is not a free variable of $B$. If $D_j$ does not depend upon $B$, then, by Proposition 2, $\Gamma \vdash D_j$ and, consequently, by Gen, $\Gamma \vdash \forall x_k D_j$. Thus, $\Gamma \vdash D_i$. Now, by axiom A1, $\vdash D_i \Rightarrow_s (B \Rightarrow_s D_i)$.
So, $\Gamma \vdash B \Rightarrow_s D_i$ by MP. If, on the other hand, $x_k$ is not a free variable of $B$, then, by axiom A5, $\vdash \forall x_k(B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s \forall x_k D_j)$ Since $\Gamma \vdash B \Rightarrow_s D_j$, we have, by Gen, $\Gamma \vdash \forall x_k(B \Rightarrow_s D_j)$, and so, by MP, $\Gamma \vdash B \Rightarrow_s \forall x_k D_j$ that is, $\Gamma \vdash B \Rightarrow_s D_i$. This completes the induction, and our proposition is just the special case $i = n$.

## Appendix B. The Generalized Recursion Theorem.

**Theorem 1.** Let $S$ be a set, $c \in S$ and $G: S \to S$ is any function with $dom(G) = S$ and $range(G) \subseteq S$. Let $W[G] \in \mathbb{N}^\# \times S$ be a binary relation such that:

(a) $(1, c) \in W[G]$ and

(b) if $(x, y) \in W[G]$ then $(Sc(x), G(y)) \in W[G]$.

Then there exists a function $F: \mathbb{N}^\# \to S$ such that:

(i) $dom(F) = \mathbb{N}^\#$ and $range(F) \subseteq S$;

(ii) $F(1) = c$;

(iii) for all $x \in \mathbb{N}^\#$, $F(Sc(x)) = G(F(x))$.

1. The desired function $F$ is a certain relation $W \subseteq \mathbb{N}^\# \times S$. It is to have the properties:

(ii') $(1, c) \in W$;

(iii') if $(x, y) \in W$ then $(Sc(x), G(y)) \in W$.

**Remark 1.** The latter is just another way of expressing (iii), that if

$$F(x) = y \qquad (B.1)$$

then

$$F(Sc(x)) = G(y). \qquad (B.2)$$

**Remark.2.** Note that any relation $W$ mentioned above is hyper inductiverelation since the hyper inductivity conditions (ii')-(iii') are satisfied.

However there are many hyper inductive relations which satisfy the conditions (ii')-(iii'); on such is $\mathbb{N}^\# \times S$. What distinguishes the desired function from all these other relations is that we want $(a, b)$ to be on it only as required by (ii') and (iii'). In other words, it is to be the smallest relation satisfying

(ii')-(iii'). This can be expressed precisely as follows:

(1) Let $M$ be a set of the relations $W$ satisfying the conditions (ii') and (iii'); then we define

$$F = \bigcap_{W \in M} W.$$

Hence

(2) whenever $W \in M$ then $F \subseteq W$.

We shall now show that we can derived from (1) that $F$ is also one relation in $M$.

(3) $(1, c) \in F$.

This follows immediately from the definition of $\bigcap_{W \in M}$ and the fact that $(1, c) \in W$ for all $W \in M$.

(4) If $(x, y) \in F$ then $(Sc(x), G(y)) \in F$.

For if $(x, y) \in F$ then $(x, y) \in W$ for all $W \in M$; hence by (iii')

$(Sc(x), G(y)) \in W$ for all $W \in M$ so that $(Sc(x), G(y)) \in F$ by (1).

We must now verify that $F$ is actually a function, i.e., we wish to show