hcju/proofwikips huggingface dataset:
Generating Function for Linearly Recurrent Sequence
Let $\sequence {a_n}$ be a linearly recurrent sequence defined as: :$a_n = \begin{cases} b_n & : 1 \le n \le m \\ c_1 a_{n - 1} + c_2 a_{n - 2} + \cdots + c_m a_{n - m} & : n > m \end{cases}$ where: :$m \in \Z_{>0}$ is a (strictly) positive integer :$b_1, \ldots, b_m$ are constants. Then the generating function for $\sequence {a_n}$ is of the form: :$\map G z = \dfrac {\map P z} {1 - c_1 z - c_2 z^2 - \cdots - c_m z^m}$ where $\map P z$ is a polynomial  in $z$ given by $b_1 z + b_2 z^2 + \cdots + b_m z^m$.
Product of Generating Functions
Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$. Let $\map H z$ be the generating function for the sequence $\sequence {b_n}$. Then $\map G z \map H z$ is the generating function for the sequence $\sequence {c_n}$, where: :$\forall n \in \Z_{\ge 0}: c_n = \displaystyle \sum_{k \mathop = 0}^n a_k b_{n - k}$
Generating Function for Sequence of Partial Sums of Series
Let $s$ be the the series: :$\displaystyle s = \sum_{n \mathop = 1}^\infty a_n = a_0 + a_1 + a_2 + a_3 + \cdots$ Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Let $\left\langle{c_n}\right\rangle$ denote the sequence of partial sums of $s$. Then the generating function for $\left\langle{c_n}\right\rangle$ is given by: :$\displaystyle \dfrac 1 {1 - z} G \left({z}\right) = \sum_{n \mathop \ge 0} c_n z^n$
Product of Generating Functions/General Rule
Let $G_0 \left({z}\right), G_1 \left({z}\right), G_2 \left({z}\right), \ldots$ be any number of generating functions (up to countably infinite) for the sequences $\left\langle{a_0 n}\right\rangle, \left\langle{a_1 n}\right\rangle, \left\langle{a_2 n}\right\rangle, \ldots$ Then: {{begin-eqn}} {{eqn | l = \prod_{j \mathop \ge 0} G_j \left({z}\right)       | r = \prod_{j \mathop \ge 0} \sum_{k \mathop \ge 0} a_{j k} z^k       | c =  }} {{eqn | r = \sum_{n \mathop \ge 0} z^n \sum_{\substack {k_0, k_1, k_2, \ldots \mathop \ge 0 \\ k_0 \mathop + k_1 \mathop + \mathop \cdots \mathop = n} } \left({\prod_{j \mathop \ge 0} a_{j k} }\right)       | c =  }} {{end-eqn}}
Product of Exponential Generating Functions
Let $G \left({z}\right)$ be the exponential generating function for the sequence $\left\langle{\dfrac {a_n} {n!} }\right\rangle$. Let $H \left({z}\right)$ be the exponential generating function for the sequence $\left\langle{\dfrac {b_n} {n!} }\right\rangle$. Then $G \left({z}\right) H \left({z}\right)$ is the generating function for the sequence $\left\langle{\dfrac {c_n} {n!} }\right\rangle$, where: :$\forall n \in \Z_{\ge 0}: c_n = \displaystyle \sum_{k \mathop \in \Z} \dbinom n k a_k b_{n - k}$
Generating Function of Multiple of Parameter
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Let $c$ be a constant. Then $G \left({c z}\right)$ be the generating function for the sequence $\left\langle{b_n}\right\rangle$ where: :$\forall n \in \Z_{\ge 0}: b_n = c^n a_n$
Generating Function for Sequence of Powers of Constant
Let $c \in \R$ be a constant. Let $\sequence {a_n}$ be the sequence defined as: :$\forall n \in \Z_{\ge 0}: a_n = c^n$ That is: :$\sequence {a_n} = 1, c, c^2, c^3, \ldots$ Then the generating function for $\sequence {a_n}$ is given as: :$\map G z = \dfrac 1 {1 - c z}$
Generating Function for Odd Terms of Sequence
Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$. Consider the subsequence $\sequence {b_n} := \tuple {a_1, a_3, a_5, \ldots}$ Then the generating function for $\sequence {b_n}$ is: :$\dfrac 1 2 \paren {\map G z - \map G {-z} }$
Derivative of Generating Function
Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$. Then: {{begin-eqn}} {{eqn | l = \frac \d {\d z} \map G z       | r = \sum_{k \mathop \ge 0} \left({k + 1}\right) a_{k + 1} z^k       | c =  }} {{eqn | r = a_1 + 2 a_2 z + 3 a_3 z^3 + \cdots       | c =  }} {{end-eqn}}
Integral of Generating Function
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Then: {{begin-eqn}} {{eqn | l = \int_0^z G \left({t}\right) \rd t       | r = \sum_{k \mathop \ge 1} \dfrac {a_{k - 1} z^k} k       | c =  }} {{eqn | r = a_0 z + \dfrac {a_1 z^2} 2 + \dfrac {a_2 z^3} 3 + \dfrac {a_3 z^4} 4 + \cdots       | c =  }} {{end-eqn}}
Generating Function for Sequence of Reciprocals of Natural Numbers
Let $\sequence {a_n}$ be the sequence defined as: :$\forall n \in \N_{> 0}: a_n = n$ That is: :$\sequence {a_n} = 1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \ldots$ Then the generating function for $\sequence {a_n}$ is given as: :$\map G z = \map \ln {\dfrac 1 {1 - z} }$
Generating Function for Sequence of Harmonic Numbers
Let $\sequence {a_n}$ be the sequence defined as: :$\forall n \in \N_{> 0}: a_n = H_n$ where $H_n$ denotes the $n$th harmonic number. That is: :$\sequence {a_n} = 1, 1 + \dfrac 1 2, 1 + \dfrac 1 2 + \dfrac 1 3, \ldots$ Then the generating function for $\sequence {a_n}$ is given as: :$\map G z = \dfrac 1 {1 - z} \map \ln {\dfrac 1 {1 - z} }$
Binomial Theorem for Negative Index and Negative Parameter
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $z \in \R$ be a real number such that $\size z < 1$. Then: {{begin-eqn}} {{eqn | l = \dfrac 1 {\paren {1 - z}^{n + 1} }       | r = \sum_{k \mathop \ge 0} \binom {-n - 1} k \paren {-z}^k       | c =  }} {{eqn | r = \sum_{k \mathop \ge 0} \binom {n + k} n z^k       | c =  }} {{end-eqn}} where $\dbinom {n + k} n$ denotes a binomial coefficient.
Power Series Expansion for Integer Power of Exponential Function minus 1
Let $e^z$ denote the exponential function. Then: {{begin-eqn}} {{eqn | l = \left({e^z - 1}\right)^n       | r = z^n + \dfrac 1 {n + 1} \left\{ { {n + 1} \atop n}\right\} z^{n + 1} + \cdots       | c =  }} {{eqn | r = n! \sum_{k \mathop \in \Z} \left\{ {k \atop n}\right\} \frac {z^k} {k!}       | c =  }} {{end-eqn}} where $\displaystyle \left\{ {k \atop n}\right\}$ denotes a Stirling number of the second kind.
Power Series Expansion for Reciprocal of 1-z to the m+1 by Logarithm of Reciprocal of 1-z
:$\dfrac 1 {\left({1 - z}\right)^{m + 1} } \ln \left({\dfrac 1 {1 - z} }\right) = \displaystyle \sum_{k \mathop \ge 1} \left({H_{m + k} - H_m}\right) \dbinom {m + k} k z^k$ where: :$\dbinom {m + k} k$ denotes a binomial coefficient :$H_m$ denotes the $m$th harmonic number.
Power Series Expansion for nth Power of Logarithm of Reciprocal of 1-z
{{begin-eqn}} {{eqn | l = \left({\ln \dfrac 1 {1 - z} }\right)^n       | r = z^n + \dfrac 1 {n + 1} \left[{ {n + 1} \atop n}\right] z^{n + 1} + \cdots       | c =  }} {{eqn | r = n! \sum_{k \mathop \in \Z} \left[{k \atop n}\right] \frac {z^k} {k!}       | c =  }} {{end-eqn}} where $\displaystyle \left[{k \atop n}\right]$ denotes an unsigned Stirling number of the first kind.
Sum over k of Stirling Numbers of Second Kind by x^k
{{begin-eqn}} {{eqn | l = \sum_k \left\{ {k \atop n}\right\} z^k       | r = \dfrac {z^n} {\prod \limits_{k \mathop = 1}^n \left({1 - k n}\right)}       | c =  }} {{eqn | r = \dfrac {z^n} {\left({1 - z}\right) \left({1 - 2 z}\right) \cdots \left({1 - n z}\right)}       | c =  }} {{end-eqn}} where: :$\displaystyle \left\{ {k \atop n}\right\}$ denotes a Stirling number of the second kind.
Sum over k of r by r+kt to the Power of k-1 over k Factorial by Power of z
Let $x$ be the continuous real function of $z$ which satisfies: :$\ln x = z x^t$ where $x = 1$ when $z = 0$. Then: {{begin-eqn}} {{eqn | l = x^r       | r = \sum_{k \mathop \ge 0} \dfrac {r \left({r + k t}\right)^{k - 1} } {k!} z^k       | c =  }} {{eqn | r = 1 + r z + \dfrac {r \left({r + 2 t}\right)} 2 z^2 + \cdots       | c =  }} {{end-eqn}}
Cauchy's Integral Formula/General Result/Corollary
Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$. Let the coefficient of $z^n$ extracted from $\map G z$ be denoted: :$\sqbrk {z^n} \map G z := a_n$ Let $\map G z$ be convergent for $z = z_0$ and $0 < r < \cmod {z_0}$. Then: :$\sqbrk {z^n} \map G z = \displaystyle \frac 1 {2 \pi i} \oint_{\cmod z \mathop = r} \dfrac {\map G z \d z} {z^{n + 1} }$
Summation of Products of n Numbers taken m at a time with Repetitions
Let $a, b \in \Z$ be integers such that $b \ge a$. Let $U$ be a set of $n = b - a + 1$ numbers $\left\{ {x_a, x_{a + 1}, \ldots, x_b}\right\}$. Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let: {{begin-eqn}} {{eqn | l = h_m       | r = \sum_{a \mathop \le j_1 \mathop \le \mathop \cdots \mathop \le j_m \mathop \le b} \paren {\prod_{k \mathop = 1}^m x_{j_k} }       | c =  }} {{eqn | r = \sum_{a \mathop \le j_1 \mathop \le \mathop \cdots \mathop \le j_m \mathop \le b} x_{j_1} \cdots x_{j_m}       | c =  }} {{end-eqn}} That is, $h_m$ is the product of all $m$-tuples of elements of $U$ taken $m$ at a time, allowing for repetition. For $r \in \Z_{> 0}$, let: :$S_r = \displaystyle \sum_{k \mathop = a}^b {x_k}^r$ Then: {{begin-eqn}} {{eqn | l = h_m       | r = \sum_{\substack {k_1, k_2, \ldots, k_m \mathop \ge 0 \\ k_1 \mathop + 2 k_2 \mathop + \mathop \cdots \mathop + m k_m \mathop = m} } \paren {\prod_{j \mathop = 1}^m \dfrac { {S_j}^{k_j} } {j^{k_j} k_j !} }       | c =  }} {{eqn | r = \sum_{\substack {k_1, k_2, \ldots, k_m \mathop \ge 0 \\ k_1 \mathop + 2 k_2 \mathop + \mathop \cdots \mathop + m k_m \mathop = m} } \dfrac { {S_1}^{k_1} } {1^{k_1} k_1 !} \dfrac { {S_2}^{k_2} } {2^{k_2} k_2 !} \cdots \dfrac { {S_m}^{k_m} } {m^{k_m} k_m !}       | c =  }} {{end-eqn}}
Derivative of Generating Function for Sequence of Harmonic Numbers
Let $\sequence {a_n}$ be the sequence defined as: :$\forall n \in \N_{> 0}: a_n = H_n$ where $H_n$ denotes the $n$th harmonic number. Let $\map G z$ be the generating function for $\sequence {a_n}$: :$\map G z = \dfrac 1 {1 - z} \map \ln {\dfrac 1 {1 - z} }$ from Generating Function for Sequence of Harmonic Numbers. Then the derivative of $\map G z$ {{WRT|Differentiation}} $z$ is given by: :$\map {G'} z = \dfrac 1 {\paren {1 - z}^2} \map \ln {\dfrac 1 {1 - z} } + \dfrac 1 {\paren {1 - z}^2}$
Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 4
:$\displaystyle \sum_{a \mathop \le j_1 \mathop \le j_2 \mathop \le j_3 \mathop \le j_4 \mathop \le b} x_{j_1} x_{j_2} x_{j_3} x_{j_4} = \dfrac { {S_1}^4} {24} + \dfrac { {S_1}^2 S_2} 4 + \dfrac { {S_2}^2} 8 + \dfrac {S_1 S_3} 3 + \dfrac {S_4} 4$ where: :$\displaystyle S_r := \sum_{k \mathop = a}^b {x_k}^r$.
Summation of Products of n Numbers taken m at a time with Repetitions/Recurrence Formula
A recurrence relation for $h_n$ can be given as: {{begin-eqn}} {{eqn | l = h_n       | r = \sum_{k \mathop = 1}^n \dfrac {S_k h_{n - k} } n       | c =  }} {{eqn | r = \dfrac 1 n \paren {S_1 h_{n - 1} + S_2 h_{n - 2} + \cdots S_n h_0}       | c =  }} {{end-eqn}} for $n \ge 1$.
Summation of Products of n Numbers taken m at a time with Repetitions/Lemma 1
Let $\map G z$ be the generating function for the sequence $\sequence {h_m}$. Then: {{begin-eqn}} {{eqn | l = \map G z       | r = \prod_{k \mathop = a}^b \dfrac 1 {1 - x_k z}       | c =  }} {{eqn | r = \dfrac 1 {\paren {1 - x_a z} \paren {1 - x_{a + 1} z} \cdots \paren {1 - x_b z} }       | c =  }} {{end-eqn}}
Summation of Products of n Numbers taken m at a time with Repetitions/Lemma 2
{{begin-eqn}} {{eqn | l = \ln \left({G \left({z}\right)}\right)       | r = \sum_{k \mathop \ge 1} \dfrac {S_k z^k} k       | c =  }} {{end-eqn}}
Generating Function for Elementary Symmetric Function
Let $U$ be a set of $n$ numbers $\set {x_1, x_2, \ldots, x_n}$. Define: {{begin-eqn}} {{eqn | l = \map {e_m} {U}       | r = \begin{cases} 1 & m = 0\\ \displaystyle \sum_{1 \mathop \le j_1 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} x_{j_1} x_{j_2} \cdots x_{j_m} & 1 \leq m \leq n \\ 0 & m \gt n \\ \end{cases}      | c = elementary symmetric function }} {{eqn | l = a_m       | r = \map {e_m} U       | c = for $m=0,1,2,\ldots$ }} {{eqn | l = \map G z       | r = \displaystyle \sum_{m=0}^\infty a_m z^m       | c = generating function for $\set {a_m}_{m=0}^\infty$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \map G z       | r = \displaystyle \prod_{k \mathop = 1}^n \paren {1 + x_k z} }} {{end-eqn}}
Newton-Girard Formulas/Lemma 1
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{h_m}\right\rangle$. Then: {{begin-eqn}} {{eqn | l = G \left({z}\right)       | r = \prod_{k \mathop = a}^b \left({1 + x_k z}\right)       | c =  }} {{eqn | r = \left({1 + x_a z}\right) \left({1 + x_{a + 1} z}\right) \cdots \left({1 + x_b z}\right)       | c =  }} {{end-eqn}}
Newton-Girard Formulas/Lemma 2
{{begin-eqn}} {{eqn | l = \ln \left({G \left({z}\right)}\right)       | r = \sum_{k \mathop \ge 1} \left({-1}\right)^{k + 1} \dfrac {S_k z^k} k       | c =  }} {{end-eqn}}
Newton-Girard Formulas/Examples/Order 3
:$\displaystyle \sum_{a \mathop \le i \mathop < j \mathop < k \mathop \le b} x_i x_j x_k = \dfrac { {S_1}^3} 6 - \dfrac {S_1 S_2} 2 + \dfrac {S_3} 3$ where: :$\displaystyle S_r := \sum_{k \mathop = a}^b {x_k}^r$
Newton-Girard Formulas/Examples/Order 4
:$\displaystyle \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < j_3 \mathop < j_4 \mathop \le b} x_{j_1} x_{j_2} x_{j_3} x_{j_4} = \dfrac { {S_1}^4} {24} - \dfrac { {S_1}^2 S_2} 4 + \dfrac { {S_2}^2} 8 + \dfrac {S_1 S_3} 3 - \dfrac {S_4} 4$ where: :$\displaystyle S_r := \sum_{k \mathop = a}^b {x_k}^r$.
Newton-Girard Formulas/Examples/Order 1
:$\displaystyle \sum_{a \mathop \le i \mathop \le b} x_i = S_1$ where: :$\displaystyle S_r := \sum_{k \mathop = a}^b {x_k}^r$
Newton-Girard Formulas/Recurrence Formula
A recurrence relation for $h_n$ can be given as: {{begin-eqn}} {{eqn | l = h_n       | r = \sum_{k \mathop = 1}^n \dfrac {\left({-1}\right)^{k + 1} S_k h_{n - k} } n       | c =  }} {{eqn | r = \dfrac 1 n \left({S_1 h_{n - 1} - S_2 h_{n - 2} + \cdots S_n h_0}\right)       | c =  }} {{end-eqn}} for $n \ge 1$.
Newton's Identities
Let $X$ be a set of $n$ numbers $\set {x_1, x_2, \ldots, x_n}$. Define: {{begin-eqn}} {{eqn | l = {\mathbf S}_m        | r = \set { \paren {j_1,\ldots,j_m} : 1 \le j_1 \lt \cdots \lt j_m \le n}       | c = $1 \le m \le n$ }} {{eqn | l = \map {e_m} {X}       | r = \begin{cases} 1 & m = 0\\ \displaystyle \sum_{ {\mathbf S}_m } x_{j_1} \cdots x_{j_m} & 1 \leq m \leq n \\ 0 & m \gt n \\ \end{cases}      | c = elementary symmetric function }} {{eqn | l = \map {p_k} X       | r = \begin{cases}                \displaystyle  n   & k = 0 \\               \displaystyle \sum_{i \mathop = 1}^n  x_i^k   & k \ge 1 \\             \end{cases}       | c = power sums }} {{end-eqn}} Then '''Newton's Identities''' are: {{begin-eqn}} {{eqn | n = 1       | l = k \, \map {e_k} X       | r = \displaystyle \sum_{i \mathop = 1}^k              \paren {-1}^{i-1}             \map {e_{k-i} } X             \map {p_i} X       | c = for $1 \leq k \leq n$ }} {{eqn | n = 2       | l = 0       | r = \displaystyle \sum_{i \mathop = k-n}^k              \paren {-1}^{i-1}             \map {e_{k-i} } X             \map {p_i} X       | c = for $1 \leq n \lt k$ }} {{end-eqn}}
Summation of Products of n Numbers taken m at a time with Repetitions/Inverse Formula
Let $S_m$ be expressed in the form: :$S_m = \displaystyle \sum_{k_1 \mathop + 2 k_2 \mathop + \mathop \cdots \mathop + m k_m \mathop = m} A_m {h_1}^{k_1} {h_2}^{k_2} \cdots {h_m}^{k_m}$ for $k_1, k_2, \ldots, k_m \ge 0$. Then : :$A_m = \left({-1}\right)^{k_1 + k_2 + \cdots + k_m - 1} \dfrac {m \left({k_1 + k_2 + \cdots + k_m - 1}\right)! } {k_1! \, k_2! \, \cdots k_m!}$
Law of Subtraction
On the following number systems: : integers $\Z$ : rational numbers $\Q$ : real numbers $\R$ : complex numbers $\C$ there exists a unique $x$ such that: :$a + x = b$ for every given $a$ and $b$. $x$ is then defined and denoted: :$x := b - a$
Law of Division
Let $\mathbb F$ denote one of the following number systems: : rational numbers $\Q$ : real numbers $\R$ : complex numbers $\C$ Let $a, b \in \mathbb F$ such that $a \ne 0$. Then there exists a unique $x$ such that: :$a x = b$ $x$ is then defined and denoted: :$x := b / a$
Limit of Rational Sequence is not necessarily Rational
Let $S = \left\langle{a_n}\right\rangle$ be a rational sequence. Let $S$ be convergent to a limit $L$. Then it is not necessarily the case that $L$ is itself a rational number.
Powers of Imaginary Unit
The (integer) powers of the imaginary unit $i$ are: {{begin-eqn}} {{eqn | l = i^0       | r = 1       | c =  }} {{eqn | l = i^1       | r = i       | c =  }} {{eqn | l = i^2       | r = -1       | c =  }} {{eqn | l = i^3       | r = -i       | c =  }} {{eqn | l = i^4       | r = 1       | c =  }} {{end-eqn}}
Real Number Multiplied by Complex Number
Let $a \in \R$ be a real number. Let $c + d i \in \C$ be a complex number. Then: :$a \times \left({c + d i}\right) = \left({c + d i}\right) \times a = a c + i a d$
Complex Number equals Negative of Conjugate iff Wholly Imaginary
Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Then $\overline z = -z$  $z$ is wholly imaginary.
Square of Complex Conjugate is Complex Conjugate of Square
Let $z \in \C$ be a complex number. Let $\overline z$ denote the complex conjugate of $z$. Then: : $\overline {z^2} = \left({\overline z}\right)^2$
Complex Modulus equals Zero iff Zero
Let $z = a + i b$ be a complex number. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod z = 0 \iff z = 0$
Complex Modulus is Non-Negative
Let $z = a + i b \in \C$ be a complex number. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod z \ge 0$
Complex Modulus equals Complex Modulus of Conjugate
Let $z \in \C$ be a complex number. Let $\overline z$ denote the complex conjugate of $z$. Let $\cmod z$ denote the modulus of $z$. Then: :$\cmod z = \cmod {\overline z}$
Square of Complex Modulus equals Complex Modulus of Square
Let $z \in \C$ be a complex number. Let $\left\vert{z}\right\vert$ be the modulus of $z$. Then: : $\left\vert{z^2}\right\vert = \left\vert{z}\right\vert^2$
Power of Complex Modulus equals Complex Modulus of Power
Let $z \in \C$ be a complex number. Let $\left\vert{z}\right\vert$ be the modulus of $z$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\left\vert{z^n}\right\vert = \left\vert{z}\right\vert^n$
Power of Complex Conjugate is Complex Conjugate of Power
Let $z \in \C$ be a complex number. Let $\overline z$ denote the complex conjugate of $z$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\overline {z^n} = \left({\overline z}\right)^n$
Conjugate of Polynomial is Polynomial of Conjugate
Let $f \left({z}\right) = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers. Let $\alpha \in \C$ be a complex number. Then: :$\overline {f \left({\alpha}\right)} = f \left({\overline \alpha}\right)$ where $\overline \alpha$ denotes the complex conjugate of $\alpha$.
Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs
Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers. Let $\alpha \in \C$ be a root of $f$. Then $\overline \alpha$ is also a root of $f$, where $\overline \alpha$ denotes the complex conjugate of $\alpha$. That is, all complex roots of $f$ appear as conjugate pairs.
Equation of Circle in Complex Plane/Formulation 1
Let $\C$ be the complex plane. Let $C$ be a circle in $\C$ whose radius is $r \in \R_{>0}$ and whose center is $\alpha \in \C$. Then $C$ may be written as: :$\cmod {z - \alpha} = r$ where $\cmod {\, \cdot \,}$ denotes complex modulus.
Equation of Circle in Complex Plane/Formulation 1/Interior
The points in $\C$ which correspond to the interior of $C$ can be defined by: :$\left\lvert{z - \alpha}\right\rvert < r$
Equation of Circle in Complex Plane/Formulation 1/Exterior
The points in $\C$ which correspond to the exterior of $C$ can be defined by: :$\left\lvert{z - \alpha}\right\rvert > r$
Condition for Complex Number to be in Right Half Plane
Let $\C$ be the complex plane. Let $P$ be the half-plane of $\C$ to the right of the infinite straight line $x = \lambda$. The points in $P$ can be defined by: :$\map \Re z > \lambda$ where $\map \Re z$ denotes the real part of $z$.
Equation of Imaginary Axis in Complex Plane
Let $\C$ be the complex plane. Let $z \in \C$ be subject to the condition: :$\cmod {z - 1} = \cmod {z + 1}$ where $\cmod {\, \cdot \,}$ denotes complex modulus. Then the locus of $z$ is the imaginary axis.
Conversion between Cartesian and Polar Coordinates in Plane
Let $S$ be the plane. Let a cartesian plane $C$ be applied to $S$. Let a polar coordinate plane $P$ be superimposed upon $C$ such that: :$(1): \quad$ The origin of $C$ coincides with the pole of $P$. :$(2): \quad$ The $x$-axis of $C$ coincides with the polar axis of $P$. Let $p$ be a point in $S$. Let $p$ be specified as $p = \polar {r, \theta}$ expressed in the polar coordinates of $P$. Then $p$ is expressed as $\tuple {r \cos \theta, r \sin \theta}$ in $C$. Contrariwise, let $p$ be expressed as $\tuple {x, y}$ in the cartesian coordinates of $C$. Then $p$ is expressed as: :$p = \polar {\sqrt {x^2 + y^2}, \arctan \dfrac y x + \pi \sqbrk {x < 0 \text{ or } y < 0} + \pi \sqbrk {x > 0 \text{ and } y < 0} }$ where: :$\sqbrk {\, \cdot \,}$ is Iverson's convention. :$\arctan$ denotes the arctangent function.
Polar Form of Reciprocal of Complex Number
Let $z := r \left({\cos \theta + i \sin \theta}\right) \in \C$ be a complex number expressed in polar form. Then: :$\dfrac 1 z = \dfrac {\cos \theta - i \sin \theta} r$
Argument of Complex Conjugate equals Argument of Reciprocal
Let $z \in \C$ be a complex number. Then: :$\arg {\overline z} = \arg \dfrac 1 z = -\arg z$ where: :$\arg$ denotes the argument of a complex number :$\overline z$ denotes the complex conjugate of $z$.
Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle
Let $z \in \C$ be a complex number. Let $z$ be interpreted as a vector in the complex plane. Let $w \in \C$ be the complex number defined as $z$ multiplied by the imaginary unit $i$: :$w = i z$ Then $w$ can be interpreted as the vector $z$ after being rotated through a right angle in an anticlockwise direction.
Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles
Let $z \in \C$ be a complex number. Let $z$ be interpreted as a vector in the complex plane. Let $w \in \C$ be the complex number defined as $z$ multiplied by $-1$: :$w = \left({-1}\right) z$ Then $w$ can be interpreted as the vector $z$ after being rotated through two right angles. The direction of rotation is usually interpreted as being anticlockwise, but a rotated through two right angles is the same whichever direction the rotation is performed.
Geometrical Interpretation of Complex Addition
Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively. Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$. Then the diagonal $OC$ of $OACB$ corresponds to $\mathbf a + \mathbf b$, the sum of $a$ and $b$ expressed as a vector.
Geometrical Interpretation of Complex Subtraction
Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively. Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$. Then the diagonal $BA$ of $OACB$ corresponds to $\mathbf a - \mathbf b$, the difference of $a$ and $b$ expressed as a vector.
Condition for Collinearity of Points in Complex Plane/Formulation 1
Let $z_1$, $z_2$ and $z_3$ be points in the complex plane. Then $z_1$, $z_2$ and $z_3$ are collinear : :$\dfrac {z_1 - z_3} {z_3 - z_2} = \lambda$ where $\lambda \in \R$ is a real number. If this is the case, then $z_3$ divides the line segment in the ratio $\lambda$. If $\lambda > 0$ then $z_3$ is between $z_1$ and $z_2$, and if $\lambda < 0$ then $z_3$ is outside the line segment joining $z_1$ to $z_2$.
Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Sum of Series
The following definition of the concept of the real exponential function: