hcju/proofwikips huggingface dataset:
Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm implies Limit of Sequence
The following definition of the concept of the real exponential function:
Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential
The following definition of the concept of the real exponential function:
Equivalence of Definitions of Real Exponential Function/Extension of Rational Exponential implies Differential Equation
The following definition of the concept of the real exponential function:
Exponential of Sum/Complex Numbers/General Result
Let $m \in \N_{>0}$ be a natural number. Let $z_1, z_2, \ldots, z_m \in \C$ be complex numbers. Let $\exp z$ be the exponential of $z$. Then: :$\displaystyle \map \exp {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$
Exponential of Sum/Complex Numbers/General Result/Corollary
Let $m \in \Z_{>0}$ be a positive integer. Let $z \in \C$ be a complex number. Let $\exp z$ be the exponential of $z$. Then: : $\displaystyle \exp \paren {m z} = \paren {\exp z}^m$
Sum over k from 1 to n of n Choose k by Sine of n Theta
:$\displaystyle \sum_{k \mathop = 1}^n \dbinom n k \sin k \theta = \paren {2 \cos \dfrac \theta 2}^n \sin \dfrac {n \theta} 2$
Point of Perpendicular Intersection on Real Line from Points in Complex Plane
Let $a, b \in \C$ be complex numbers represented by the points $A$ and $B$ respectively in the complex plane. Let $x \in \R$ be a real number represented by the point $X$ on the real axis such that $AXB$ is a right triangle with $X$ as the right angle. Then: :$x = \dfrac {a_x - b_x \pm \sqrt {a_x^2 + b_x^2 + 2 a_x b_x - 4 a_y b_y} } 2$ where: : $a = a_x + a_y i, b = b_x + b_y i$ {{mistake|What rubbish. Working on it.}}
Condition for Points in Complex Plane to form Parallelogram/Examples/2+i, 3+2i, 2+3i, 1+2i
The points in the complex plane represented by the complex numbers: :$2 + i, 3 + 2 i, 2 + 3 i, 1 + 2 i$ are the vertices of a square.
Equation of Ellipse in Complex Plane
Let $\C$ be the complex plane. Let $E$ be an ellipse in $\C$ whose major axis is $d \in \R_{>0}$ and whose foci are at $\alpha, \beta \in \C$. Then $C$ may be written as: :$\cmod {z - \alpha} + \cmod {z - \beta} = d$ where $\cmod {\, \cdot \,}$ denotes complex modulus.
Condition for Collinearity of Points in Complex Plane/Formulation 2
Let $z_1, z_2, z_3$ be distinct complex numbers. Then: : $z_1, z_2, z_3$ are collinear in the complex plane : ::$\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 = 0$ :where: ::$\alpha + \beta + \gamma = 0$ ::not all of $\alpha, \beta, \gamma$ are zero.
Quadrilateral in Complex Plane is Cyclic iff Cross Ratio of Vertices is Real
Let $z_1, z_2, z_3, z_4$ be distinct complex numbers. Then: : $z_1, z_2, z_3, z_4$ define the vertices of a cyclic quadrilateral  their cross ratio: : $\paren {z_1, z_3; z_2, z_4} = \dfrac {\paren {z_1 - z_2} \paren {z_3 - z_4} } {\paren {z_1 - z_4} \paren {z_3 - z_2} }$ is wholly real.
Definition:Cross Ratio
Let $z_1, z_2, z_3, z_4$ be distinct complex numbers. The cross ratio of $z_1, z_2, z_3, z_4$ is defined and denoted: : $\paren {z_1, z_3; z_2, z_4} = \dfrac {\paren {z_1 - z_2} \paren {z_3 - z_4} } {\paren {z_1 - z_4} \paren {z_3 - z_2} }$
Equation relating Points of Parallelogram in Complex Plane
Let $ABVU$ be a parallelogram in the complex plane whose vertices correspond to the complex numbers $a, b, v, u$ respectively. Let $\angle BAU = \alpha$. Let $\cmod {UA} = \lambda \cmod {AB}$. :510px Then: :$u = \paren {1 - q} a + q b$ :$v = -q a + \paren {1 + q} b$ where: :$q = \lambda e^{i \alpha}$
Circle of Apollonius in Complex Plane
Let $\C$ be the complex plane. Let $\lambda \in \R$ be a real number such that $\lambda \ne 0$ and $\lambda \ne 1$. Let $a, b \in \C$ such that $a \ne b$. The equation: :$\cmod {\dfrac {z - a} {z - b} } = \lambda$ decribes a circle of Apollonius $C$ in $\C$ such that: :if $\lambda < 0$, then $a$ is inside $C$ and $b$ is outside :if $\lambda > 0$, then $b$ is inside $C$ and $a$ is outside. If $\lambda = 1$ then $z$ describes the perpendicular bisector of the line segment joining $a$ to $b$.
P-adic Norm is Non-Archimedean Norm
The $p$-adic norm forms a non-Archimedean norm on the rational numbers $\Q$.
Circle of Apollonius is Circle
Let $A, B$ be distinct points in the plane. Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $X$ be the locus of points in the plane such that: :$XA = \lambda \paren {XB}$ Then $X$ is in the form of a circle, known as a circle of Apollonius. :400px If $\lambda < 1$, then $A$ is inside the circle, and $B$ is outside. If $\lambda > 1$, then $B$ is inside the circle, and $A$ is outside.
Cosine of Angle plus Integer Multiple of Pi
:$\map \cos {\theta + n \pi} = \paren {-1}^n \cos \theta$
Sine of Angle plus Integer Multiple of Pi
:$\map \sin {\theta + n \pi} = \paren {-1}^n \sin \theta$
Complex Division/Examples/(1 + sin theta + i cos theta) (1 + sin theta - i cos theta)^-1
:$\dfrac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta} = \sin \theta + i \cos \theta$
Sum of 1 + sin pi by 5 plus i cos pi by 5 to Fifth Power plus i times its Conjugate
:$\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 = 0$
Complex Roots of Unity include 1
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity. Then $1 \in U_n$. That is, $1$ is always one of the complex $n$th roots of unity of any $n$.
Positive Real Complex Root of Unity
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \in U_n$ such that $x \in \R_{>0}$ is: :$x = 1$ That is, $1$ is the only complex $n$th root of unity which is a positive real number.
Real Complex Roots of Unity for Even Index
Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is even. Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \in U_n$ such that $x \in \R$ are: :$x = 1$ or $x \in -1$ That is, $1$ and $-1$ are the only complex $n$th roots of unity which are real number.
Modulus of Complex Root of Unity equals 1
Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is even. Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity. Let $z \in U_n$. Then: :$\cmod z = 1$ where $\cmod z$ denotes the modulus of $z$.
First Complex Root of Unity is Primitive
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \set {z \in \C: z^n = 1}$ Let $\alpha_1 = \exp \paren {\dfrac {2 \pi i} n}$ denote the first complex root of unity. Then $\alpha_1$ is a primitive complex root of unity.
Powers of Primitive Complex Root of Unity form Complete Set
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \set {z \in \C: z^n = 1}$ Let $\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$ denote the $k$th complex root of unity. Let $\alpha_k$ be a primitive complex root of unity. Let $V_k = \set { {\alpha_k}^r: r \in \set {0, 1, \ldots, n - 1} }$. Then: : $V_k = U_n$ That is, $V_k = \set { {\alpha_k}^r: r \in \set {0, 1, \ldots, n - 1} }$ forms the complete set of complex $n$th roots of unity.
Sum of Powers of Primitive Complex Roots of Unity
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \set {z \in \C: z^n = 1}$ Let $\alpha = \exp \paren {\dfrac {2 k \pi i} n}$ denote a primitive complex $n$th root of unity. Let $s \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^{n - 1} \alpha^{j s}       | r = 1 + \alpha^s + \alpha^{2 s} + \cdots + \alpha^{\paren {n - 1} s}       | c =  }} {{eqn | r = \begin {cases} n & : n \divides s \\ 0 & : n \nmid s \end {cases}       | c =  }} {{end-eqn}} where: :$n \divides s$ denotes that $n$ is a divisor of $s$ :$n \nmid s$ denotes that $n$ is not a divisor of $s$.
Difference of Two Powers/Examples/Difference of Two Cubes/Corollary
:$x^3 - 1 = \paren {x - 1} \paren {x^2 + x + 1}$
Sum of Cube Roots of Unity
Let $U_3 = \set {1, \omega, \omega^2}$ denote the Cube Roots of Unity. Then: :$1 + \omega + \omega^2 = 0$
Sum of Cubes of Three Indeterminates Minus 3 Times their Product
For indeterminates $x, y, z$: :$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$ where $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$
Combination Theorem for Cauchy Sequences/Sum Rule
:$\sequence {x_n + y_n}$ is a Cauchy sequence.
Power of Complex Number minus 1
Let $z \in \C$ be a complex number. Then: :$z^n - 1 = \displaystyle \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k}$ where $\alpha$ is a primitive complex $n$th root of unity.
Power of Complex Number minus 1/Corollary
Let $z \in \C$ be a complex number. Then: :$\displaystyle \sum_{k \mathop = 0}^{n - 1} z^k = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}$ where $\alpha$ is a primitive complex $n$th root of unity.
Product of Differences between 1 and Complex Roots of Unity
Let $\alpha$ be a primitive complex $n$th root of unity. Then: :$\displaystyle \prod_{k \mathop = 1}^{n - 1} \paren {1 - \alpha^k} = n$
Complex Roots of Unity occur in Conjugate Pairs
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \set {z \in \C: z^n = 1}$ Let $\alpha \in U_n$ be the first complex $n$th root of unity. Then: :$\forall k \in \Z_{>0}, k < \dfrac n 2: \overline {\alpha^k} = \alpha^{n - k}$ That is, each of the complex $n$th roots of unity occur in conjugate pairs: :$\tuple {\alpha, \alpha^{n - 1} }; \tuple {\alpha^2, \alpha^{n - 2} }; \ldots; \tuple {\alpha^s, \alpha^{n - s} }$ where: :$s = \dfrac {n - 1} 2$ for odd $n$ :$s = \dfrac {n - 2} 2$ for even $n$.
Combination Theorem for Cauchy Sequences/Product Rule
:$\sequence {x_n y_n}$ is a Cauchy sequence.
Factorisation of x^(2n+1)-1 in Real Domain
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n + 1} - 1 = \paren {z - 1} \displaystyle \prod_{k \mathop = 1}^n \paren {z^2 - 2 \cos \dfrac {2 \pi k} {2 n + 1} + 1}$
Factorisation of z^n-a
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $a \in \C$ be a complex number. Then: :$z^n - a = \displaystyle \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k b}$ where: :$\alpha$ is a primitive complex $n$th root of unity :$b$ is any complex number such that $b^n = a$.
Cosine of 144 Degrees
:$\cos 144 \degrees = \cos \dfrac {4 \pi} 5 = -\dfrac \phi 2 = -\dfrac {1 + \sqrt 5} 4$
Sine of 144 Degrees
:$\sin 144 \degrees = \cos \dfrac {4 \pi} 5 = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$
Cube Root of Unity if Modulus is 1 and Real Part is Minus Half
Let $z \in \C$ be a complex number such that: :$\cmod z = 1$ :$\Re \paren z = -\dfrac 1 2$ where: :$\cmod z$ denotes the complex modulus of $z$ :$\Re \paren z$ denotes the real part of $z$. Then: :$z^3 = 1$
Sum of Two Cubes in Complex Domain
:$a^3 + b^3 = \paren {a + b} \paren {a \omega + b \omega^2} \paren {a \omega^2 + b \omega}$ where: : $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$
Roots of Complex Number/Examples/z^8 + 1 = 0
The roots of the polynomial: :$z^8 + 1 = 0$ are: :$\set {\cos \dfrac {\paren {2 k + 1} \pi} 8 + i \sin \dfrac {\paren {2 k + 1} \pi} 8: k \in \set {0, 1, \ldots, 7} }$
Quadruple Angle Formulas/Cosine/Factor Form
:$\cos 4 \theta = \paren {\cos \theta - \cos \dfrac \pi 8} \paren {\cos \theta - \cos \dfrac {3 \pi} 8} \paren {\cos \theta - \cos \dfrac {5 \pi} 8} \paren {\cos \theta - \cos \dfrac {7 \pi} 8}$
Factorisation of z^n+1
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$z^n + 1 = \displaystyle \prod_{k \mathop = 0}^{n - 1} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} n}$
Factorisation of z^(2n)+1 in Real Domain
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n} + 1 = \displaystyle \prod_{k \mathop = 1}^n \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} {2 n} + 1}$
Roots of Complex Number/Examples/z^5 + 1 = 0
The roots of the polynomial: :$z^5 + 1 = 0$ are: :$\set {\cos \dfrac \pi 5 \pm i \sin \dfrac \pi 5, \cos \dfrac {3 \pi} 5 \pm i \sin \dfrac {3 \pi} 5, -1}$
Combination Theorem for Cauchy Sequences
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $\sequence {x_n}$,  $\sequence {y_n}$ be Cauchy sequences in $R$. Let $a, b \in R$. The following results hold:
Convergence of Modulus of Convergent Complex Sequence
Let $\sequence {z_n}$ be a sequence in $\C$. Let $\sequence {z_n}$ converge to a value $c \in \C$. Let $\cmod z$ denote the modulus of a complex number $z$. Then: :$\sequence {\cmod {z_n} }$ converges to a value $\cmod c$.
Combination Theorem for Sequences/Real/Difference Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$
Combination Theorem for Sequences/Complex/Sum Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
Combination Theorem for Sequences/Complex
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$. Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} z_n = c$ :$\displaystyle \lim_{n \mathop \to \infty} w_n = d$ Let $\lambda, \mu \in \C$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Complex/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Complex/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Complex/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Complex/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Complex/Product Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Complex/Quotient Rule}}
Combination Theorem for Sequences/Complex/Difference Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$
Combination Theorem for Sequences/Complex/Multiple Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
Combination Theorem for Sequences/Complex/Combined Sum Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$
Combination Theorem for Sequences/Normed Division Ring/Sum Rule
:$\sequence {x_n + y_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
Absolute Value of Product
Let $x, y \in \R$ be real numbers. Then: :$\size {x y} = \size x \size y$ where $\size x$ denotes the absolute value of $x$.
Combination Theorem for Sequences/Complex/Product Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$
Combination Theorem for Sequences/Complex/Quotient Rule
:$\displaystyle \lim_{n \mathop \to \infty} \frac {z_n} {w_n} = \frac c d$ provided that $d \ne 0$.
Combination Theorem for Sequences/Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$,  $\sequence {y_n} $ be sequences in $R$. Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ :$\displaystyle \lim_{n \mathop \to \infty} y_n = m$ Let $\lambda, \mu \in R$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Product Rule}} === Inverse Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Inverse Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Quotient Rule}}
Combination Theorem for Sequences/Normed Division Ring/Product Rule
:$\sequence {x_n y_n}$ is convergent to the limit $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$
Combination Theorem for Sequences/Normed Division Ring/Multiple Rule
:$\sequence {\lambda x_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule
:$\sequence {\lambda x_n + \mu y_n }$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
Combination Theorem for Sequences/Normed Division Ring/Difference Rule
:$\sequence {x_n - y_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$
Cauchy's Convergence Criterion for Series
A series $\displaystyle \sum_{i \mathop = 0}^\infty a_i$ is convergent  for every $\epsilon > 0$ there is a number $N \in \N$ such that: :$\size {a_{n + 1} + a_{n + 2} + \cdots + a_m} < \epsilon$ holds for all $n \ge N$ and $m > n$. {{explain|What domain is $\sequence {a_n}$ in?}}
Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $R$. Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ :$\displaystyle \lim_{n \mathop \to \infty} y_n = m$ Then: {{:Combination Theorem for Sequences/Normed Division Ring/Product Rule}}
Convergent Sequence in Normed Division Ring is Bounded
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Then $\sequence {x_n}$ is bounded.
Metric Induced by Norm on Normed Division Ring is Metric
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by $\norm{\,\cdot\,}$. Then $d$ is a metric.
Existence of Radius of Convergence of Complex Power Series/Divergence
Let ${B_R}^- \paren \xi$ denote the closed $R$-ball of $\xi$. Let $z \notin {B_R}^- \paren \xi$. Then $S \paren z$ is divergent.
Radius of Convergence of Power Series in Complex Plane
Consider the complex power series: :$S = \displaystyle \sum_{k \mathop = 0}^\infty z^n$ The radius of convergence $S$ is $1$.
Radius of Convergence of Power Series Expansion for Cosine Function
The cosine function has the complex power series expansion: {{begin-eqn}} {{eqn | l = C \paren z       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots       | c =  }} {{end-eqn}} which is the power series expansion of the cosine function. This is valid for all $z \in \C$.
Radius of Convergence of Power Series Expansion for Sine Function
The sine function has the complex power series expansion: {{begin-eqn}} {{eqn | l = S \paren z       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots       | c =  }} {{end-eqn}} which is the power series expansion of the sine function. This is valid for all $z \in \C$.
Product of Absolutely Convergent Series
Let $f \paren z = \displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $g \paren z = \displaystyle \sum_{n \mathop = 1}^\infty b_n$ be two real or complex series that are absolutely convergent. Then $f \paren z g \paren z$ is an absolutely convergent series, and: :$f \paren z g \paren z = \displaystyle \sum_{n \mathop = 1}^\infty c_n$ where: :$c_n = \displaystyle \sum_{k \mathop = 1}^n a_k b_{n - k}$
Sum of Infinite Series of Product of nth Power of Cosine by nth Multiple of Cosine
Let $0 < \theta < \dfrac \pi 2$. Then: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty \cos^n \theta \, \map \cos {n + 1} \theta       | r = \cos \theta + \cos \theta \cos 2 \theta + \cos^2 \theta \cos 3 \theta + \cos^3 \theta \cos 4 \theta + \cdots       | c =  }} {{eqn | r = 0       | c =  }} {{end-eqn}}
Euler's Formula/Real Domain
Let $\theta \in \R$ be a real number. Then: :$e^{i \theta} = \cos \theta + i \sin \theta$
Reverse Triangle Inequality/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Then: :$\forall x, y \in R: \norm {x - y} \ge \bigsize {\norm x - \norm y}$
Sequence Converges to Within Half Limit/Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l \ne 0$ Then: :$\exists N: \forall n > N: \norm {x_n} > \dfrac {\norm l} 2$
Limit of Subsequence equals Limit of Sequence/Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\sequence {x_{n_r} }$ is convergent and $\displaystyle \lim_{r \mathop \to \infty} x_{n_r} = l$
Combination Theorem for Sequences/Normed Division Ring/Quotient Rule
Suppose $m \ne 0$. Then: :$\exists k \in \N : \forall n \in \N: y_{k + n} \ne 0$ and the sequences: :$\sequence {x_{k + n} \ {y_{k + n} }^{-1} }$ and $\sequence { {y_{k + n} }^{-1} \ x_{k + n} }$ are well-defined and convergent with: :$\displaystyle \lim_{n \mathop \to \infty} x_{k + n} \ {y_{k + n} }^{-1} = l m^{-1}$ :$\displaystyle \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} \ x_{k + n} = m^{-1} l$
Boundedness of Real Sine and Cosine
Let $x \in \R$ be a real number. Then:
Real Sine Function is Bounded
:$\size {\sin x} \le 1$
Complex Cosine Function is Unbounded
The complex cosine function is unbounded.
Argument of Exponential is Imaginary Part plus Multiple of 2 Pi
Let $z \in \C$ be a complex number. Let $\exp z$ denote the complex exponential of $z$. Let $\arg z$ denote the argument of $z$. Then: :$\map \arg {\exp z} = \set {\Im z + 2 k \pi: k \in \Z}$ where $\Im z$ denotes the imaginary part of $z$.
Real Part of Sine of Complex Number
Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\sin z$ denote the complex sine function. Then: :$\map \Re {\sin z} = \sin x \cosh y$ where: :$\Re z$ denotes the real part of a complex number $z$ :$\sin$ denotes the sine function (real and complex) :$\cosh$ denotes the hyperbolic cosine function.
Inverse Tangent of i
The inverse tangent of $i$ is not defined.
Cauchy-Hadamard Theorem/Complex Case
Let $\xi \in \C$ be a complex number. Let $\displaystyle S \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a (complex) power series about $\xi$. Then the radius of convergence $R$ of $S \paren z$ is given by: :$\displaystyle \dfrac 1 R = \limsup_{n \mathop \to \infty} \cmod {a_n}^{1/n}$ If: :$\displaystyle \limsup_{n \mathop \to \infty} \cmod {a_n}^{1/n} = 0$ then the radius of convergence is infinite, and $S \paren z$ is absolutely convergent for all $z \in \C$.
Cauchy-Hadamard Theorem/Real Case
Let $\xi \in \R$ be a real number. Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$. Then the radius of convergence $R$ of $S \paren x$ is given by: :$\displaystyle \frac 1 R = \limsup_{n \mathop \to \infty} \size {a_n}^{1/n}$ If: :$\displaystyle \frac 1 R = \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = 0$ then the radius of convergence is infinite and therefore the interval of convergence is $\R$.