Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial

{{begin-eqn}} {{eqn | l = n!       | r = \sum_{k \mathop \ge 0} \dfrac { {!k} \, n^{\underline k} } {k!}       | c =  }} {{eqn | r = \dfrac { !0 \times n^{\underline 0} } {0!} + \dfrac { {!1} \times n^{\underline 1} } {1!} + \dfrac { {!2} \times n^{\underline 2} } {2!} + \dfrac { {!3} \times n^{\underline 3} } {3!} + \cdots       | c =  }} {{eqn | r = 1 + \left({1 - \dfrac 1 {1 !} }\right) n + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} }\right) n \left({n - 1}\right) + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} - \dfrac 1 {3 !} }\right) n \left({n - 1}\right) \left({n - 2}\right) + \cdots       | c =  }} {{end-eqn}}


Factorial of Half

:$\left({\dfrac 1 2}\right)! = \dfrac {\sqrt \pi} 2$


Laplace Transform of Complex Power

Let $q$ be a constant complex number with $\map \Re q > -1$ Let $t^q: \R_{>0} \to \C$ be a branch of the complex power multifunction chosen such that $f$ is continuous on the half-plane $\map \Re s > 0$. Then $f$ has a Laplace transform given by: :$\laptrans {t^q} = \dfrac {\map \Gamma {q + 1} } {s^{q + 1} }$ where $\Gamma$ denotes the gamma function.


Local Minimum of Gamma Function on Positive Domain

The local minimum of the Gamma function on the positive real numbers occurs at the point: :$\left({1 \cdotp 46163 21449 68362 34126 26595, 0 \cdotp 88560 31944 10888 70027 88159}\right)$ {{OEIS|A030169|order = $x$-coordinate}} {{OEIS|A030171|order = $y$-coordinate}}


Properties of Falling Factorial

Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$. This page gathers together some of the properties of the falling factorial.


Properties of Rising Factorial

Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$. This page gathers together some of the properties of the rising factorial.


Integer to Power of Itself Less One Falling is Factorial

:$n^{\underline {n - 1} } = n!$


Recursive Form of Generalized Termial

The termial function as defined on the real numbers fulfils the identity: :$x? = x + \left({x - 1}\right)?$


Euler Form of Gamma Function at Positive Integers

The Euler form of the Gamma function: :$\displaystyle \Gamma \left({z}\right) := \lim_{m \mathop \to \infty} \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$ converges to the factorial function at positive integers: :$\displaystyle \lim_{m \mathop \to \infty} \frac {m^n m!} {\left({n + 1}\right) \left({n + 2}\right) \cdots \left({n + m}\right)} = n!$


Gamma Function of Minus One Half

:$\map \Gamma {-\dfrac 1 2} = -2 \sqrt \pi$


Diameter of N-Cube

Let $Q_n = \closedint {c - R} {c + R}^n$ be an $n$-cube in Euclidean $n$-Space equipped with the usual metric. Then the diameter of $Q_n$ is given by: :$\map {\operatorname {diam} } {Q_n} = 2 R \sqrt n$


Legendre's Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime number. Let $n$ be expressed in base $p$ representation. Let $r$ be the digit sum of the representation of $n$ in base $p$. Then $n!$ is divisible by $p^\mu$ but not by $p^{\mu + 1}$, where: :$\mu = \dfrac {n - r} {p - 1}$


Uniform Convergence of General Dirichlet Series

Let $\arg \left({z}\right)$ denote the argument of the complex number $z \in \C$. Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n \left({s}\right)}$ be a general Dirichlet series. Let $f \left({s}\right)$ converge at $s_0 = \sigma_0 + i t_0$. Then $f \left({s}\right)$ converges uniformly for all $s$ such that: :$\left\vert{\arg \left({s - s_0}\right)}\right\vert \le a < \dfrac \pi 2$


Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Condition for Convergence

Consider the series: {{:Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial}} This converges only when $n \in \Z_{\ge 0}$, that is, when $n$ is a non-negative integer.


Bounded Subspace of Euclidean Space is Totally Bounded

Let $\struct {\R^n, \norm {\, \cdot \,} }$ be a Euclidean space, where $\norm {\, \cdot \,}$ denotes the usual metric. Let $M$ be a metric subspace of $\struct {\R^n, \norm {\, \cdot \,} }$. Let $M$ be bounded. Then $M$ is totally bounded.


Difference is Rational is Equivalence Relation

Define $\sim$ as the relation on real numbers given by: :$x \sim y \iff x - y \in \Q$ That is, that the difference between $x$ and $y$ is rational. Then $\sim$ is an equivalence relation.


Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta

:$\displaystyle \prod_{n \mathop \ge 1} \dfrac {\left({n + \alpha_1}\right) \cdots \left({n + \alpha_k}\right)} {\left({n + \beta_1}\right) \cdots \left({n + \beta_k}\right)} = \dfrac {\Gamma \left({1 + \beta_1}\right) \cdots\Gamma \left({1 + \beta_1}\right)} {\Gamma \left({1 + \alpha_1}\right) \cdots\Gamma \left({1 + \alpha_k}\right)}$ where: :$\alpha_1 + \cdots + \alpha_k = \beta_1 + \cdots + \beta_k$ :none of the $\beta$s is a negative integer.


Characterization of N-Cube

Let $\struct {\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$. Let $x, y \in \R^n$, where: :$x = \tuple {x_1, x_2, \ldots, x_n}$ :$y = \tuple {y_1, y_2, \ldots, y_n}$ Let $R > 0$ be fixed. Let: :$\displaystyle Q = \set {x, y \in \R^n: \sup_{x, y} \max_i \size {y_i - x_i} \le R}$ Then $Q$ is an $n$-cube.


Factorial of Integer plus Reciprocal of Integer

Let $x \in \Z$ be a positive integer. Then: :$\ds \lim_{n \mathop \to \infty} \dfrac {\paren {n + x}!} {n! n^x} = 1$


Lower and Upper Bound of Factorial

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$\dfrac {n^n} {e^{n - 1} } \le n! \le \dfrac {n^{n + 1} } {e^{n - 1} }$


Sum of Indices of Falling Factorial

:$x^{\underline {m + n} } = x^{\underline m} \left({x - m}\right)^{\underline n}$


Sum of Indices of Rising Factorial

:$x^{\overline {m + n} } = x^{\overline m} \paren {x + m}^{\overline n}$


Union of Indexed Family of Sets Equal to Union of Disjoint Sets

Let $\family {E_n}_{n \mathop \in \N}$ be a countable indexed family of sets where at least two $E_n$ are distinct. Then there exists a countable indexed family of disjoint sets $\family {F_n}_{n \mathop \in \N}$ defined by: :$\displaystyle F_k = E_k \setminus \paren {\bigcup_{j \mathop = 0}^{k \mathop - 1} E_j}$ satisfying: :$\displaystyle \bigsqcup_{n \mathop \in \N} F_n = \bigcup_{n \mathop \in \N} E_n$ where $\bigsqcup$ denotes disjoint union.


Borel Sigma-Algebra Generated by Closed Sets

Let $\mathcal B \left({S, \tau}\right)$ be a Borel $\sigma$-algebra generated by the set of open sets in $S$. Then $\mathcal B \left({S, \tau}\right)$ is equivalently generated by the set of closed sets in $S$.


Stirling Number of n with n-m is Polynomial in n of Degree 2m/Unsigned First Kind

Let $m \in \Z_{\ge 0}$. The unsigned Stirling number of the first kind $\displaystyle \left[{n \atop n - m}\right]$ is a polynomial in $n$ of degree $2 m$.


Unsigned Stirling Number of the First Kind of n with n-2

:$\displaystyle \left[{n \atop n - 2}\right] = \binom n 4 + 2 \binom {n + 1} 4$


Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum

Let $\MM$ be an infinite $\sigma$-algebra on a set $X$. Then $\MM$ is has cardinality at least that of the cardinality of the continuum $\mathfrak c$: :$\map \Card \MM \ge \mathfrak c$


Stirling Number of the Second Kind of n with n-2

:$\displaystyle {n \brace n - 2} = \binom {n + 1} 4 + 2 \binom n 4$


Unsigned Stirling Number of the First Kind of n with n-3

:$\displaystyle \left[{n \atop n - 3}\right] = \binom n 6 + 8 \binom {n + 1} 6 + 6 \binom {n + 2} 6$


Stirling Number of the Second Kind of n with n-3

:$\displaystyle {n \brace n - 3} = \binom {n + 2} 6 + 8 \binom {n + 1} 6 + 6 \binom n 6$


Duality Law for Stirling Numbers

For all integers $n, m \in \Z$: :$\displaystyle \left\{ {n \atop m}\right\} = \left[{-m \atop -n}\right]$ where: :$\displaystyle \left\{ {n \atop m}\right\}$ denotes a Stirling number of the second kind :$\displaystyle \left[{n \atop m}\right]$ denotes an unsigned Stirling number of the first kind.


Power of Complex Number as Summation of Stirling Numbers of Second Kind

Let $z \in \C$ be a complex number whose real part is positive. Then: :$z^r = \displaystyle \sum_{k \mathop \in \Z} {r \brace r - k} z^{\underline {r - k} }$ where: :$\displaystyle {r \brace r - k}$ denotes the extension of the Stirling numbers of the second kind to the complex plane :$z^{\underline {r - k} }$ denotes $z$ to the $r - k$ falling.


Non-Divisbility of Binomial Coefficients of n by Prime

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $p$ be a prime number. Then: :$\dbinom n k$ is not divisible by $p$ for any $k \in \Z_{\ge 0}$ where $0 \le k \le n$ {{iff}}: :$n = a p^m - 1$ where $1 \le a < p$ for some $m \in \Z_{\ge 0}$.


Summation Formula for Reciprocal of Binomial Coefficient

{{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 0} \binom n k \dfrac {\paren {-1}^k} {k + x}       | r = \dfrac 1 {x \binom {n + x} n}       | c =  }} {{eqn | r = \dfrac {n!} {x \paren {x + 1} \cdots \paren {x + n} }       | c =  }} {{end-eqn}} as long as the denominators are not zero.


Sum over k of r Choose k by Minus r Choose m Minus 2k

Let $r \in \R$, $m \in \Z$. :$\displaystyle \sum_{k \mathop \in \Z} \binom r k \binom {-r} {m - 2 k} \paren {-1}^{m + k} = \binom r m$


Sigma-Algebra Contains Generated Sigma-Algebra of Subset

Let  $\sigma_\FF$ be a  be a $\sigma$-algebra on a set $\FF$. Let $\sigma_\FF$ contain a set of sets $\EE$. Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$. Then $\map \sigma \EE \subseteq \sigma_\FF$


Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset

Let $\sigma\left({\mathcal F}\right)$ be the $\sigma$-algebra generated by $\mathcal E$. Let $\sigma\left({\mathcal F}\right)$ contain a set of sets $\mathcal E$. Let $\sigma \left({\mathcal E}\right)$ be the $\sigma$-algebra generated by $\mathcal E$. Then $\sigma \left({\mathcal E}\right) \subseteq \sigma\left({\mathcal F}\right)$.


Binomial Theorem/Abel's Generalisation/x+y = 0

Consider Abel's Generalisation of Binomial Theorem: {{:Abel's Generalisation of Binomial Theorem}} This holds in the special case where $x + y = 0$.


Binomial Theorem/Hurwitz's Generalisation

:$\displaystyle \paren {x + y}^n = \sum x \paren {x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}^{\epsilon_1 + \cdots + \epsilon_n - 1} \paren {y - \epsilon_1 z_1 - \cdots - \epsilon_n z_n}^{n - \epsilon_1 - \cdots - \epsilon_n}$ where the summation ranges over all $2^n$ choices of $\epsilon_1, \ldots, \epsilon_n = 0$ or $1$ independently.


Set of Relations can be Ordered by Inclusion

Let $S \times T$ be the product of two sets. Let $\mathcal R$ be a set of relations on $S \times T$. Then $\mathcal R$ can be ordered by inclusion.


Set of Mappings can be Ordered by Inclusion

Let $S \times T$ be the product of two sets. Let $\mathcal F$ be a set of mappings on $S \times T$. Then $\mathcal F$ can be ordered by inclusion.


Binomial Theorem/Abel's Generalisation/Negative n

Abel's Generalisation of Binomial Theorem: {{:Abel's Generalisation of Binomial Theorem}} does not hold for $n \in \Z_{< 0}$.


Summation from k to m of r Choose k by s Choose n-k by nr-(r+s)k

:$\displaystyle \sum_{k \mathop = 0}^m \dbinom r k \dbinom s {n - k} \paren {n r - \paren {r + s} k} = \paren {m + 1} \paren {n - m} \dbinom r {m + 1} \dbinom s {n - m}$


Factors of Binomial Coefficient/Corollary 2

For all $r \in \R, k \in \Z$: :$\dbinom r {k - 1} = k \paren {r - k} \dbinom r k$


Summation from k to m of 2k-1 Choose k by 2n-2k Choose n-k by -1 over 2k-1

:$\displaystyle \sum_{k \mathop = 0}^m \binom {2 k - 1} k \binom {2 n - 2 k} {n - k} \dfrac {-1} {2 k - 1} = \dfrac {n - m} {2 n} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m} + \dfrac 1 2 \dbinom {2 n} n$


Inverse of Pascal's Triangle expressed as Matrix

Consider Pascal's triangle expressed as a (square) matrix $\mathbf M$, with the top left element holding $\dbinom 0 0$. :$\begin{pmatrix} 1 &  0 &  0 &   0 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  1 &  0 &   0 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  2 &  1 &   0 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  3 &  3 &   1 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  4 &  6 &   4 &   1 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  5 & 10 &  10 &   5 &   1 &   0 &   0 &   0 & \cdots \\ 1 &  6 & 15 &  20 &  15 &   6 &   1 &   0 &   0 & \cdots \\ 1 &  7 & 21 &  35 &  35 &  21 &   7 &   1 &   0 & \cdots \\ 1 &  8 & 28 &  56 &  70 &  56 &  28 &   8 &   1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$ The inverse $\mathbf M^{-1}$ of $\mathbf M$ is the same as $\mathbf M$ but with alternate elements negated, starting with the elements below the main diagonal: :$\begin{pmatrix}  1 &  0 &   0 &   0 &   0 &   0 &  0 &  0 & 0 & \cdots \\ -1 &  1 &   0 &   0 &   0 &   0 &  0 &  0 & 0 & \cdots \\  1 & -2 &   1 &   0 &   0 &   0 &  0 &  0 & 0 & \cdots \\ -1 &  3 &  -3 &   1 &   0 &   0 &  0 &  0 & 0 & \cdots \\  1 & -4 &   6 &  -4 &   1 &   0 &  0 &  0 & 0 & \cdots \\ -1 &  5 & -10 &  10 &  -5 &   1 &  0 &  0 & 0 & \cdots \\  1 & -6 &  15 & -20 &  15 &  -6 &  1 &  0 & 0 & \cdots \\ -1 &  7 & -21 &  35 & -35 &  21 & -7 &  1 & 0 & \cdots \\  1 & -8 &  28 & -56 &  70 & -56 & 28 & -8 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$


Binomial Coefficient of Half

Let $k \in \Z$. :$\dbinom {\frac 1 2} k = \dfrac {\left({-1}\right)^{k - 1} } {4^k \left({2 k - 1}\right)} \dbinom {2 k} k$ where $\dbinom {\frac 1 2} k$ denotes a binomial coefficient.


Binomial Coefficient of Half/Corollary

Let $k \in \Z_{\ge 0}$. :$\dbinom {\frac 1 2} k = \dfrac {\left({-1}\right)^{k - 1} } {2^{2 k - 1} \left({2 k - 1}\right)} \dbinom {2 k - 1} k - \delta_{k 0}$ where $\dbinom {\frac 1 2} k$ denotes a binomial coefficient.


Wosets are Isomorphic to Each Other or Initial Segments

Let $\struct {S, \preceq_S}$ and $\struct {T, \preceq_T}$ be well-ordered sets. Then precisely one of the following hold: :$\struct {S, \preceq_S}$ is order isomorphic to $\struct {T, \preceq_T}$ or: :$\struct {S, \preceq_S}$ is order isomorphic to an initial segment in $\struct {T, \preceq_T}$ or: :$\struct {T, \preceq_T}$ is order isomorphic to an initial segment in $\struct {S, \preceq_S}$


Gaussian Binomial Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \sum_{k \mathop \in \Z} \dbinom n k_q q^{k \paren {k - 1} / 2} x^k       | r = \prod_{k \mathop = 1}^n \paren {1 + q^{k - 1} x}       | c =  }} {{eqn | r = \paren {1 + x} \paren {1 + q x} \paren {1 + q^2 x} \cdots \paren {1 + q^{n - 1} x}       | c =  }} {{end-eqn}} where $\dbinom n k_q$ denotes a Gaussian binomial coefficient.


Addition Rule for Gaussian Binomial Coefficients/Formulation 1

:$\dbinom n m_q = \dbinom {n - 1} m_q + \dbinom {n - 1} {m - 1}_q q^{n - m}$


Addition Rule for Gaussian Binomial Coefficients

Let $\dbinom n m_q$ denote a Gaussian binomial coefficient.


Union of Initial Segments is Initial Segment or All of Woset

Let $\struct {X, \preccurlyeq}$ be a well-ordered non-empty set. Let $A \subseteq X$. Let: :$\ds J = \bigcup_{x \mathop \in A} S_x$ be a union of initial segments defined by the elements of $A$. Then either: :$J = X$ or: :$J$ is an initial segment of $X$.


Countable Subset of Minimal Uncountable Well-Ordered Set Has Upper Bound

Let $\Omega$ denote the minimal uncountable well-ordered set. Let $\omega$ be a countable subset of $\Omega$. Then $\omega$ has an upper bound in $\Omega$.


Gaussian Binomial Coefficient of 1

:$\dbinom 1 m_q = \delta_{0 m} + \delta_{1 m}$ That is: :$\dbinom 1 m_q = \begin{cases} 1 & : m = 0 \text { or } m = 1 \\ 0 & : \text{otherwise} \end{cases}$ where $\dbinom 1 m_q$ denotes a Gaussian binomial coefficient.


Symmetry Rule for Gaussian Binomial Coefficients

Let $q \in \R_{\ne 1}, n \in \Z_{>0}, k \in \Z$. Then: :$\dbinom n k_q = \dbinom n {n - k}_q$ where $\dbinom n k_q$ is a Gaussian binomial coefficient.


Gaussian Binomial Theorem/Real Numbers

Let $r \in \R$ be a real number. :$\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \left({k - 1}\right) / 2} x^k = \prod_{k \mathop \ge 0} \dfrac {1 + q^k x} {1 + q^{r + k} x}$ where: :$\dbinom r k_q$ denotes a Gaussian binomial coefficient. :$x \in \R: \left\lvert{x}\right\rvert < 1$ :$q \in \R: \left\lvert{q}\right\rvert < 1$.


Gaussian Binomial Theorem/Negation of Upper Index

Let $r \in \R$ be a real number. {{begin-eqn}} {{eqn | l = \prod_{k \mathop \ge 0} \dfrac {1 + q^{k + r + 1} x} {1 + q^k x}       | r = \sum_{k \mathop \in \Z} \dbinom {-r - 1} k_q q^{k \left({k - 1}\right) / 2} \left({-q^{r + 1} x}\right)^k       | c =  }} {{eqn | r = \sum_{k \mathop \in \Z} \dbinom {k + r} x^k       | c =  }} {{end-eqn}} where: :$\dbinom r k_q$ denotes a Gaussian binomial coefficient. :$x \in \R: \left\lvert{x}\right\rvert < 1$ :$q \in \R: \left\lvert{q}\right\rvert < 1$.


Cardinality of Set of Combinations with Repetition

Let $S$ be a finite set with $n$ elements The number of $k$-combinations of $S$ with repetition is given by: :$N = \dbinom {n + k - 1} k$


Sum over k of Unsigned Stirling Number of the First Kind of n+1 with k+1 by Stirling Number of the Second Kind of k with m by -1^k-m

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k {n + 1 \brack k + 1} {k \brace m} \paren {-1}^{k - m} = \sqbrk {n \ge m} \dfrac {n!} {m!}$ where: :$\sqbrk {n \ge m}$ is Iverson's convention :$\displaystyle {n + 1 \brack k + 1}$ denotes an unsigned Stirling number of the first kind :$\displaystyle {k \brace m}$ denotes a Stirling number of the second kind.


Dixon's Identity/General Case

For $l, m, n \in \Z_{\ge 0}$: :$\displaystyle \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\paren {l + m + n}!} {l! \, m! \, n!}$


Dixon's Identity

For $n \in \Z_{\ge 0}$: :$\displaystyle \sum_{k \mathop \in \Z} \paren {-1}^k \binom {2 n} {n + k}^3 = \dfrac {\paren {3 n}!} {\paren {n!}^3}$


Dixon's Identity/Gaussian Binomial Form

=== Formulation 1 === {{:Dixon's Identity/Gaussian Binomial Form/Formulation 1}} === Formulation 2 === {{:Dixon's Identity/Gaussian Binomial Form/Formulation 2}}


Dixon's Identity/Gaussian Binomial Form/Formulation 1

For $l, m, n \in \Z_{\ge 0}$: :$\displaystyle \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {m - r - s} k_q \dbinom {n + r - s} {n - k}_q \dbinom {r + k} {m + n}_q = \dbinom r m_q \dbinom s n_q$ where $\dbinom r m_q$ denotes a Gaussian binomial coefficient


Sum over j, k of -1^j+k by j+j Choose k+l by r Choose j by n Choose k by s+n-j-k Choose m-j

Let $l, m, n \in \Z$ be integers such that $n \ge 0$. Then: :$\displaystyle \sum_{j, k \mathop \in \Z} \left({-1}\right)^{j + k} \dbinom {j + k} {k + l} \dbinom r j \dbinom n k \dbinom {s + n - j - k} {m - j} = \left({-1}\right)^l \dbinom {n + r} {n + l} \dbinom {s - r} {m - n - l}$


Analytic Continuation of Generating Function of Dirichlet Series

Let $\displaystyle \map \lambda s = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^s}$ be a Dirichlet series  Let $c \in \R$ be greater than the abscissa of absolute convergence of $\lambda$ and greater than $0$. Let $\displaystyle \map g z = \sum_{k \mathop = 1}^\infty \map \lambda {c k} z^k $ be the generating function of $\map \lambda {c k}$ Then the analytic continuation of $g$ to $\C$ is equal to  :$\displaystyle \sum_{n \mathop = 1}^\infty a_n  \frac z {n^c - z}$


Uniqueness of Real z such that x Choose n+1 Equals y Choose n+1 Plus z Choose n

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $x, y \in \R$ be real numbers which satisfy: :$n \le y \le x \le y + 1$ Then there exists a unique real number $z$ such that: :$\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$ where $n - 1 \le z \le y$.


X Choose n leq y Choose n + z Choose n-1 where n leq y leq x leq y+1 and n-1 leq z leq y

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $x, y \in \R$ be real numbers which satisfy: :$n \le y \le x \le y + 1$ Let $z$ be the unique real number $z$ such that: :$\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$ where $n - 1 \le z \le y$. Its uniqueness is proved at Uniqueness of Real $z$ such that $\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$. Then: :$\dbinom x n \le \dbinom y n + \dbinom z {n - 1}$


Upper Bound for Binomial Coefficient

Let $n, k \in \Z$ such that $n \ge k \ge 0$. Then: :$\dbinom n k \le \left({\dfrac {n e} k}\right)^k$ where $\dbinom n k$ denotes a binomial coefficient.


Lower Bound for Binomial Coefficient

Let $n, k \in \Z$ such that $n \ge k \ge 0$. Then: :$\dbinom n k \ge \left({\dfrac {\left({n - k}\right) e} k}\right)^k \dfrac 1 {e k}$ where $\dbinom n k$ denotes a binomial coefficient.


Sum over k of n Choose k by p^k by (1-p)^n-k by Absolute Value of k-np

Let $n \in \Z_{\ge 0}$ be a non-negative integer. Then: :$\displaystyle \sum_{k \mathop \in \Z} \dbinom n k p^k \left({1 - p}\right)^{n - k} \left\lvert{k - n p}\right\rvert = 2 \left\lceil{n p}\right\rceil \dbinom n {\left\lceil{n p}\right\rceil} p^{\left\lceil{n p}\right\rceil} \left({1 - p}\right)^{n - 1 - \left\lceil{n p}\right\rceil}$


Sequence of General Harmonic Numbers Converges for Index Greater than 1

Let $H_n^{\paren r}$ denote the general harmonic number: :$\displaystyle H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$ for $r \in \R_{>0}$. Let $r > 1$. Then as $n \to \infty$, $H_n^{\paren r}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.


Generating Function of Bernoulli Polynomials

Let $\map {B_n} x$ denote the $n$th Bernoulli polynomial. Then the generating function for $B_n$ is: :$\displaystyle \frac {t e^{t x} } {e^t - 1} = \sum_{k \mathop = 0}^\infty \frac {\map {B_k} x} {k!} t^k$


Symmetry of Bernoulli Polynomial

Let $\map {B_n} x$ denote the nth Bernoulli polynomial. Then: :$\map {B_n} {1 - x} = \paren {-1}^n \map {B_n} x$


Value of Odd Bernoulli Polynomial at One Half

Let $B_n \left({x}\right)$ denote the $n$th Bernoulli polynomial. Then: : $B_{2 n + 1} \left({\dfrac 1 2}\right) = 0$


Harmonic Number is Greater than Logarithm plus Gamma

:$H_n > \ln n + \gamma$ where: :$H_n$ denotes the $n$th harmonic number :$\gamma$ denotes the Euler-Mascheroni constant.


Summation to n of Power of k over k

:$\displaystyle \sum_{k \mathop = 1}^n \dfrac {x^k} k = H_n + \displaystyle \sum_{k \mathop = 1}^n \dbinom n k \dfrac {\left({x - 1}\right)^k} k$ where: :$H_n$ denotes the $n$th harmonic number :$\dbinom n k$ denotes a binomial coefficient.


Harmonic Number as Unsigned Stirling Number of First Kind over Factorial

:$H_n = \dfrac {\left[{ {n + 1} \atop 2}\right]} {n!}$ where: :$H_n$ denotes the $n$th harmonic number :$n!$ denotes the $n$th factorial :$\displaystyle \left[{ {n + 1} \atop 2}\right]$ denotes an unsigned Stirling number of the first kind.


Summation over k to n of Natural Logarithm of k

:$\ds \sum_{k \mathop = 1}^n \ln k = \map \ln {n!}$ where $n!$ denotes the $n$th factorial.


Difference between Summation of Natural Logarithms and Summation of Harmonic Numbers

:$\displaystyle \sum_{k \mathop = 1}^n H_k - \sum_{k \mathop = 1}^n \map \ln {n!} \approx \gamma n + \dfrac {\ln n} 2 + 0 \cdotp 158$ where: :$H_k$ denotes the $k$th harmonic number :$n!$ denotes the $n$th factorial :$\gamma$ denotes the Euler-Mascheroni constant.


Summation to n of kth Harmonic Number over k

:$\displaystyle \sum_{k \mathop = 1}^n \dfrac {H_k} k = \dfrac { {H_n}^2 + H_n^{\paren 2} } 2$ where: :$H_n$ denotes the $n$th harmonic number :$H_n^{\paren 2}$ denotes a general harmonic number.


Principle of Recursive Definition for Well-Ordered Sets

Let $J$ be a well-ordered set. Let $C$ be any set. Let $\mathcal F$ be the set of all functions that map initial segments $S_a$ of $J$ into $C$. Then for any function of the form: :$\rho: \mathcal F \to C$ there exists a unique function: :$h: J \to C$ satisfying: :$\forall \alpha \in J: h\left({\alpha}\right) = \rho\left({ h {\restriction_{S_\alpha}} }\right)$ where ${\restriction}$ denotes the restriction of a mapping.


Angles with Parallel or Perpendicular Arms are Equal or Supplementary

Let $A$ and $B$ be angles such that their corresponding arms are either parallel or perpendicular. Then $A$ and $B$ are either equal or supplementary.


Sum of External Angles of Polygon equals Four Right Angles

Let the external angles of a polygon be generated in the same direction going around the polygon. Then the sum of all these external angles equals $4$ right angles.


Bisector of Apex of Isosceles Triangle also Bisects Base

Let $\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $AD$ be the bisector of $\angle BAC$ such that $AD$ intersects $BC$ at $D$. Then $AD$ bisects $BC$.


Bisector of Apex of Isosceles Triangle is Perpendicular to Base

Let $\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $AD$ be the bisector of $\angle BAC$ such that $AD$ intersects $BC$ at $D$. Then $AD$ is perpendicular to $BC$.


Equilateral Triangle is Equiangular

Let $\triangle ABC$ be an equilateral triangle. Then $\triangle ABC$ is also equiangular.


Equiangular Triangle is Equilateral

Let $\triangle ABC$ be equiangular. Then $\triangle ABC$ is an equilateral triangle.


No Order Isomophism Between Distinct Initial Segments of Woset

Let $E$ be a well-ordered set. Let $S_\alpha, S_\beta$ be initial segments of $E$ that are order isomorphic. Then $S_\alpha = S_\beta$.


Perpendicular is Shortest Straight Line from Point to Straight Line

Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$. Then the length of $CD$ is less than the length of all other line segments that can be drawn from $C$ to $AB$.


Straight Lines which make Equal Angles with Perpendicular to Straight Line are Equal

Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$. Let $E, F$ be points on $AB$ such that $\angle DCE = \angle DCF$. Then $CE = CF$.


Straight Line making Larger Angle with Perpendicular to Straight Line is Longer

Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$. Let $E, F$ be points on $AB$ such that $\angle DCE > \angle DCF$. Then $CE > CF$.


Two Equal Straight Lines can be Constructed from Point to Straight Line

Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Then exactly $2$ straight lines $CD$ and $CE$ can be drawn such that $CD = CE$ and $D, E$ on $AB$.


Diagonals of Rhombus Bisect Each Other at Right Angles

Let $ABCD$ be a rhombus. The diagonals $AC$ and $BD$ of $ABCD$ bisect each other at right angles.


Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel

Let $ABCD$ be a quadrilateral. Then: :$ABCD$ is a parallelogram {{iff}}: :$AB = CD$ and $AB \parallel CD$ where $AB \parallel CD$ denotes that $AB$ is parallel to $CD$.


Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel

Let $ABCD$ be a quadrilateral. Then: :$ABCD$ is a parallelogram {{iff}}: :either $AB = CD$ and $AD = BC$ :or $AB \parallel CD$ and $AD \parallel BC$ where $AB \parallel CD$ denotes that $AB$ is parallel to $CD$.


Quadrilateral is Parallelogram iff Both Pairs of Opposite Angles are Equal

Let $ABCD$ be a quadrilateral. Then: :$ABCD$ is a parallelogram {{iff}}: :$\angle ABC = \angle ADC$ and $\angle BAD = \angle BCD$.


Quadrilateral is Parallelogram iff Diagonals Bisect each other

Let $ABCD$ be a quadrilateral. Then: : $ABCD$ is a parallelogram {{iff}}: :both: :: $AD$ is a bisector of $BC$ :and: :: $BC$ is a bisector of $AD$.


Parallelograms are Congruent if Two Adjacent Sides and Included Angle are respectively Equal

Let $ABCD$ and $EFGH$ be parallelograms. Then $ABCD$ and $EFGH$ are congruent if: : $2$ adjacent sides of $ABCD$ are equal to $2$ corresponding adjacent sides of $EFGH$ : the angle between those $2$ adjacent sides on both $ABCD$ and $EFGH$ are equal.


Parallel Lines which intercept Equal Segments on Transversals

Let $3$ or more parallel lines intersect equal line segments on one transversal. Then those same $3$ or more parallel lines intersect equal line segments on every transversal.


Line Parallel to Side of Triangle which Bisects One Side also Bisects Other Side

Let $ABC$ be a triangle. Let $DE$ be a straight line parallel to $BC$. Let $DE$ bisect $AB$. Then $DE$ also bisects $AC$. That is, $DE$ is a midline of $\triangle ABC$. 400px


Midline Theorem

The midline of a triangle is parallel to the third side of that triangle and half its length.


Midline and Median of Triangle Bisect Each Other

Let $\triangle ABC$ be a triangle. Let $DE$ be the midline of $\triangle ABC$ which bisects $AB$ and $AC$. Let $AF$ be the median of $ABC$ which bisects $BC$. Then $AF$ and $DE$ bisect each other.


Diagonals of Rectangle are Equal

The diagonals of a rectangle are equal.


Summation of Odd Reciprocals in terms of Harmonic Numbers

:$\displaystyle \sum_{k \mathop = 1}^n \dfrac 1 {2 k - 1} = H_{2 n} - \dfrac {H_n} 2$ where $H_n$ denotes the $n$th harmonic number.


Numerator of p-1th Harmonic Number is Divisible by Prime p

Let $p$ be an odd prime. Consider the harmonic number $H_{p - 1}$ expressed in canonical form. The numerator of $H_{p - 1}$ is divisible by $p$.


Numerator of p-1th Harmonic Number is Divisible by p^2 for Prime Greater than 3

Let $p$ be a prime number such that $p > 3$. Consider the harmonic number $H_{p - 1}$ expressed in canonical form. The numerator of $H_{p - 1}$ is divisible by $p^2$.


Summation of Power Series by Harmonic Sequence

Consider the power series: :$\map f x = \displaystyle \sum_{k \mathop \ge 0} a_k x^k$ Let $\map f x$ converge for $x = x_0$. Then: :$\displaystyle \sum_{k \mathop \ge 0} a_k {x_0}^k H_k = \int_0^1 \dfrac {\map f {x_0} - \map f {x_0 y} } {1 - y} \rd y$ where $H_n$ denotes the $n$th harmonic number.


Summation over k to n of Harmonic Numbers over n+1-k

:$\displaystyle \sum_{k \mathop = 1}^n \dfrac {H_k} {n + 1 - k} = {H_{n + 1} }^2 - H_{n + 1}^{\paren 2}$ where $H_k$ denotes the $k$th harmonic number.


Summation over k to n of Harmonic Number k by Harmonic Number n-k

:$\displaystyle \sum_{k \mathop = 1}^n H_k H_{n - k} = \paren {n + 1} \paren { {H_n}^2 - H_n^{\paren 2} } - 2 n \paren {n_n - 1}$ where $H_k$ denotes the $k$th harmonic number.


Extension of Harmonic Number to Non-Integer Argument

Let $\map H x$ be the real function defined as: :$\map H x = \gamma + \dfrac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }$ where: :$\gamma$ denotes the Euler-Mascheroni constant :$\Gamma$ denotes the gamma function :$\Gamma'$ denotes the derivative of the gamma function. Then $H$ is an extension of the mapping $H: \N \to \Q$ defined as: :$\forall n \in \N: \map H n = H_n$ where $H_n$ denotes the $n$th harmonic number.


Initial Segment Determined by Smallest Element is Empty

Let $\left({S, \preceq}\right)$ be a well-ordered set, where $S$ is non-empty. Let $s_0 = \min S$, the smallest element of $S$. Then the initial segment determined by $s_0$, $S_{s_0}$, is empty.


Empty Mapping is Injective

Let $\nu: \varnothing \to T$ be an empty mapping. Then $\nu$ is an injection.


Empty Mapping to Empty Set is Bijective

Let $\nu: \varnothing \to \varnothing$ be an empty mapping. Then $\nu$ is a bijection.


Upper and Lower Bound of Fibonacci Number

For all $n \in \N_{> 0}$: :$\phi^{n - 2} \le F_n \le \phi^{n - 1}$ where: :$F_n$ is the $n$th Fibonacci number :$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$


Generating Function for Fibonacci Numbers

Let $\map G z$ be the function defined as: :$\map G z = \dfrac z {1 - z - z^2}$ Then $\map G z$ is a generating function for the Fibonacci numbers.


Summation over k to n of Product of kth with n-kth Fibonacci Numbers

:$\displaystyle \sum_{k \mathop = 0}^n F_k F_{n - k} = \dfrac {\left({n - 1}\right) F_n + 2n F_{n - 1} } 5$ where $F_n$ denotes the $n$th Fibonacci number.


Fibonacci Number in terms of Smaller Fibonacci Numbers/Negative Indices

Let $n \in \Z_{< 0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then Fibonacci Number in terms of Smaller Fibonacci Numbers: :$F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$ continues to hold, whether $m$ or $n$ are positive or negative.


Euler-Binet Formula/Negative Index

Let $n \in \Z_{< 0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then the Euler-Binet Formula: :$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$ continues to hold.


Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less

Let $n \in \Z$. Then: :$\phi^n = F_n \phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\phi$ denotes the golden mean.


Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less/Positive Index

Let $n \in \Z_{\ge 0}$. Then: :$\phi^n = F_n \phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\phi$ denotes the golden mean.


Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less/Negative Index

Let $n \in \Z_{\le 0}$. Then: :$\phi^n = F_n \phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number as extended to negative indices :$\phi$ denotes the golden mean.


Fibonacci Number by One Minus Golden Mean plus Fibonacci Number of Index One Less

Let $n \in \Z$. Then: :$\hat \phi^n = F_n \hat \phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\hat \phi$ denotes the $1$ minus the golden mean: ::$\hat \phi := 1 - \phi$


Reciprocal Form of One Minus Golden Mean

:$\hat \phi = - \dfrac 1 \phi$ where: :$\phi$ denotes the golden mean :$\hat \phi$ denotes one minus the golden mean: $\hat \phi = 1 - \phi$.


Closed Form of One Minus Golden Mean

:$\hat \phi = \dfrac {1 - \sqrt 5} 2$ where: :$\hat \phi$ denotes one minus the golden mean: $\hat \phi = 1 - \phi$.


Second Order Fibonacci Number in terms of Fibonacci Numbers

The second order Fibonacci number $\mathcal F_n$ can be expressed in terms of Fibonacci numbers as: :$\dfrac {3 n + 3} 5 F_n - \dfrac n 5 F_{n + 1}$


General Fibonacci Number in terms of Fibonacci Numbers

Let $r$ and $s$ be numbers, usually integers but not necessarily so limited. Let $\left\langle{a_n}\right\rangle$ be the general Fibonacci sequence : :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ a_{n - 2} + a_{n - 1} & : n > 1 \end{cases}$ Then $a_n$ can be expressed in Fibonacci numbers as: :$a_n = F_{n - 1} r + F_n s$


Fibonacci Number plus Constant in terms of Fibonacci Numbers

Let $c$ be a number. Let $\left\langle{b_n}\right\rangle$ be the sequence defined as: :$b_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ b_{n - 2} + b_{n - 1} + c & : n > 1 \end{cases}$ Then $\left\langle{b_n}\right\rangle$  can be expressed in Fibonacci numbers as: :$b_n = c F_{n - 1} + \left({c + 1}\right) F_n - c$


Fibonacci Number plus Binomial Coefficient in terms of Fibonacci Numbers

Let $m \in \Z_{>0}$ be a positive integer. Let $\left\langle{a_n}\right\rangle$ be the sequence defined as: :$a_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ a_{n - 2} + a_{n - 1} + \dbinom {n - 2} m & : n > 1 \end{cases}$ where $\dbinom {n - 2} m$ denotes a binonial coefficient. Then $\left\langle{a_n}\right\rangle$  can be expressed in Fibonacci numbers as: :$a_n = F_{m + 1} F_{n - 1} + \left({F_{m + 2} + 1}\right) F_n - \displaystyle \sum_{k \mathop = 0}^m \dbinom {n + m - k} k$


Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers

Let $f \left({n}\right)$ and $g \left({n}\right)$ be arbitrary arithmetic functions. Let $\left\langle{a_n}\right\rangle$ be the sequence defined as: :$a_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ a_{n - 1} + a_{n - 2} + f \left({n - 2}\right) & : n > 1 \end{cases}$ Let $\left\langle{b_n}\right\rangle$ be the sequence defined as: :$b_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ b_{n - 1} + b_{n - 2} + g \left({n - 2}\right) & : n > 1 \end{cases}$ Let $\left\langle{c_n}\right\rangle$ be the sequence defined as: :$c_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ c_{n - 1} + c_{n - 2} + x f \left({n - 2}\right) + y g \left({n - 2}\right) & : n > 1 \end{cases}$ where $x$ and $y$ are arbitrary. Then $\left\langle{c_n}\right\rangle$ can be expressed in Fibonacci numbers as: :$c_n = x a_n + y b_n + \left({1 - x - y}\right) F_n$


Vajda's Identity/Formulation 1

:$F_{n + i} F_{n + j} - F_n F_{n + i + j} = \paren {-1}^n F_i F_j$


Vajda's Identity/Formulation 2

:$F_{n + k} F_{m - k} - F_n F_m = \left({-1}\right)^n F_{m - n - k} F_k$


Sum of Squares of Consecutive Fibonacci Numbers

:${F_n}^2 + {F_{n + 1} }^2 = F_{2 n + 1}$ where $F_n$ denotes the $n$th Fibonacci number.


Hausdorff Maximal Principle implies Well-Ordering Theorem

Let the Hausdorff Maximal Principle hold. Then the Well-Ordering Theorem holds.


Cosine of 36 Degrees

:$\cos 36 \degrees = \cos \dfrac \pi 5 = \dfrac \phi 2 = \dfrac {1 + \sqrt 5} 4$ where $\phi$ denotes the golden mean.


Sine of 36 Degrees

:$\sin 36^\circ = \sin \dfrac \pi 5 = \dfrac {\sqrt {\sqrt 5 / \phi} } 2 = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$ where $\phi$ denotes the golden mean.


Partial Sums of Power Series with Fibonacci Coefficients

:$\displaystyle \sum_{k \mathop = 0}^n F_k x^k = \begin{cases} \dfrac {x^{n + 1} F_{n + 1} + x^{n + 2} F_n - x} {x^2 + x - 1} & : x^2 + x - 1 \ne 0 \\ \dfrac {\left({n + 1}\right) x^n F_{n + 1} + \left({n + 2}\right) x^{n + 1} F_n - 1} {2 x + 1} & : x^2 + x - 1 = 0 \end{cases}$ where $F_n$ denotes the $n$th Fibonacci number.


Sum over k of n Choose k by Fibonacci t to the k by Fibonacci t-1 to the n-k by Fibonacci m+k

:$\displaystyle \sum_{k \mathop \ge 0} \binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$ where: :$\dbinom n k$ denotes a binomial coefficient :$F_n$ denotes the $n$th Fibonacci number.


Matrix whose Determinant is Fibonacci Number

The $n \times n$ determinant: :$D_n = \begin{vmatrix} 1 & -1 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & -1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1 & 1 & -1 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 1 & 1 & -1 \\ 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 1 \\ \end{vmatrix}$ evaluates to $F_{n + 1}$.


Determinant with Unit Element in Otherwise Zero Column

Let $D$ be the determinant: :$D = \begin{vmatrix}   1 & b_{12} & \cdots & b_{1n} \\   0 & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\   0 & b_{n2} & \cdots & b_{nn} \end{vmatrix}$ Then: :$D = \begin{vmatrix}   b_{22} & \cdots & b_{2n} \\ \vdots & \ddots & \vdots \\   b_{n2} & \cdots & b_{nn} \end{vmatrix}$


Well-Ordering Theorem implies Hausdorff Maximal Principle

Let the Well-Ordering Theorem hold. Then the Hausdorff Maximal Principle holds.


Fibonacci Number with Prime Index 2n+1 is Congruent to 5^n Modulo p

Let $p = 2 n + 1$ be an odd prime. Then: :$F_p \equiv 5^n \pmod p$


Fibonacci Number n+1 Minus Golden Mean by Fibonacci Number n

:$F_{n + 1} - \phi F_n = \hat \phi^n$ where: :$F_n$ denotes the $n$th Fibonacci number :$\phi$ denotes the golden mean.


Recurrence Relation where n+1th Term is A by nth term + B to the n

Let $\left\langle{a_n}\right\rangle$ be the sequence defined by the recurrence relation: :$a_n = \begin{cases} 0 & : n = 0 \\ A a_{n - 1} + B^{n - 1} & : n > 0 \end{cases}$ for numbers $A$ and $B$. Then the closed form for $\left\langle{a_n}\right\rangle$ is given by: :$a_n = \begin{cases} \dfrac {A^n - B^n} {A - B} & : A \ne B \\ n A^{n - 1} & : A = B \end{cases}$


Fibonomial Coefficient is Integer

Let $\dbinom n k_\mathcal F$ be a Fibonomial coefficient. Then $\dbinom n k_\mathcal F$ is an integer.


Recurrence Relation for Fibonomial Coefficients

:$\dbinom n k_\mathcal F = F_{k - 1} \dbinom {n - 1} k_\mathcal F + F_{n - k + 1} \dbinom {n - 1} {k - 1}_\mathcal F$ where: :$\dbinom n k_\mathcal F$ denotes a Fibonomial coefficient :$F_{k - 1}$ etc. denote Fibonacci numbers.


Golden Mean by One Minus Golden Mean equals Minus 1

:$\phi \hat \phi = -1$ where: :$\phi$ denotes the golden mean :$\hat \phi := 1 - \phi$


Remainder of Fibonacci Number Divided by Fibonacci Number is Plus or Minus Fibonacci Number

Let $F_n$ and $F_m$ be Fibonacci numbers. By the Division Theorem, let: :$F_n = q F_m + r$ where: :$q \in \Z$ :$r \in \Z: 0 \le r < \size {F_m}$ Then either $r$ or $\size {F_m} - r$, or both, is a Fibonacci number.


Residue of Fibonacci Number Modulo Fibonacci Number

Let $F_n$ denote the $n$th Fibonacci number. Let $m, r$ be non-negative integers. Then: :$F_{m n + r} \equiv \paren {\begin{cases} F_r & : m \bmod 4 = 0 \\ \paren {-1}^{r + 1} F_{n - r} & : m \bmod 4 = 1 \\ \paren {-1}^n F_r & : m \bmod 4 = 2 \\ \paren {-1}^{r + 1 + n} F_{n - r} & : m \bmod 4 = 3 \end{cases} } \pmod {F_n}$


Sine of Multiple of Pi by 2 plus i by Natural Logarithm of Golden Mean

Let $z = \dfrac \pi 2 + i \ln \phi$. Then: :$\dfrac {\sin n z} {\sin z} = i^{1 - n} F_n$ where: :$\phi$ denotes the golden mean :$F_n$ denotes the $n$th Fibonacci number.


Golden Mean as Root of Quadratic

The golden mean $\phi$ is one of the roots of the quadratic equation: :$x^2 - x - 1 = 0$ The other root is $\hat \phi = 1 - \phi$.


Proper Subtower is Initial Segment

Let $\left({T_1,\preccurlyeq}\right)$ be a proper subtower of $\left({T_2,\preccurlyeq}\right)$. Then $\left({T_1,\preccurlyeq}\right)$ is an initial segment of $\left({T_2,\preccurlyeq}\right)$.


Power of Golden Mean as Sum of Smaller Powers

Let $\phi$ denote the golden mean. Then: :$\forall z \in \C: \phi^z = \phi^{z - 1} + \phi^{z - 2}$


100 in Golden Mean Number System is Equivalent to 011

Consider the golden mean number system. Let $p$ and $q$ be arbitrary strings in $\left\{ {0, 1}\right\}$. Let $x \in \R_{\ge 0}$ have a representation which includes the string $100$, say: :$x = p100q$ Then $x \in \R_{\ge 0}$ also has the representation: :$x = p011q$ Similarly, let $x \in \R_{\ge 0}$ have a representation which includes the string $011$, say: :$x = p011q$ Then $x \in \R_{\ge 0}$ also has the representation: :$x = p100q$ That is, any instance of $100$ appearing in a representation of a non-negative real number $x$ is equivalent to $011$, and vice versa. Note that the instance of $100$ or $011$ may also include a radix point; the instance of $011$ or $100$ to which it is equivalent will include the radix point in the same location.


Conversion of Number in Golden Mean Number System to Simplest Form

Let $x \in \R_{\ge 0}$ have a representation $S$ in the golden mean number system. Then $S$ can be converted to its simplest form as follows: :$(1): \quad$ Replace any infinite string on the right hand end of $S$ of the form $010101 \ldots$ with $100$ :$(2): \quad$ Repeatedly replace the leftmost instance of $011$ with $100$.


Simplest Form of Non-Negative Number in Golden Mean Number System is Unique

Let $x \in \R_{\ge 0}$ be represented in the golden mean number system. Let $S$ be the representation for $x$ in its simplest form. Then $S$ is unique in the sense that there exists no other representation of $x$ in simplest form.


Incidence of Double Letters in Fibonacci String

Let $S_n$ denote the $n$th Fibonacci string. Then: :$(1):\quad$ There are no instances of $2$ $\text a$'s together :$(2):\quad$ There are no instances of $3$ $\text b$'s together in $S_n$.


Fibonacci String Begins with ba

Let $S_n$ be a Fibonacci string of length $n$. Then for $n \ge 3$, $S_n$ begins with $\text{ba}$.


Fibonacci String Ends with ab or ba

Let $S_n$ be a Fibonacci string of length $n$. Then for $n \ge 3$, $S_n$ ends either with $\text{ba}$ or with $\text{ab}$.


Count of a's and b's in Fibonacci String

Let $S_n$ denote the $n$th Fibonacci string. Then for $n \ge 3$, $S_n$ has: :$F_{n - 2}$ instances of $\text a$ :$F_{n - 1}$ instances of $\text b$.


Initial Part of Fibonacci String

Let $n \in \Z_{>1}$. Let $S_n$ denote the $n$th Fibonacci string. Let $m \in \Z$ such that $1 < m \le n$. Let $F_m$ denote the $m$th Fibonacci number. The initial part of $S_n$ of length $F_m$ is the Fibonacci string $S_m$.


Positions of Instances of a in Fibonacci String

Let $S_n$ denote the $n$th Fibonacci string. Let $F_n$ denote the $n$th Fibonacci number. Let $m \in \Z$ such that $m \le F_n$. Let $m - 1$ be expressed in Zeckendorf representation as $Z_{m - 1}$. Then the $m$th letter of $S_n$ is $\text a$ {{iff}}: : $k_r = 2$ where $k_r$ denotes the final digit of $Z_{m - 1}$.


Positions of Instances of b in Fibonacci String

Let $S_n$ denote the $n$th Fibonacci string. Let $F_n$ denote the $n$th Fibonacci number. Let $k \in \Z$ such that $k \le F_n$. Then the $k$th letter of $S_n$ is $\text b$ {{iff}}: : $\left\lfloor{\left({k + 1}\right) \phi^{-1} }\right\rfloor - \left\lfloor{k \phi^{-1} }\right\rfloor = 1$ where: : $\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function : $\phi$ denotes the golden mean.


Optimal Strategy for Fibonacci Nim

Consider a game of Fibonacci nim with $n$ counters. Let it be the turn of player $\text A$. Let the maximum number of counters that can be taken by $\text A$ be $q$. Let $n$ be expressed in Zeckendorf representation as: :$n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$ Then $\text A$ can force a win {{iff}}: :$F_{k_r} \le q$ and by taking those $F_{k_r}$ counters.


Recursively Defined Sequence/Examples/Term is Term of Index less 1 plus 6 times Term of Index less 2

Consider the integer sequence $\left\langle{a_n}\right\rangle$ defined recursively as: :$a_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ a_{n - 1} + 6 a_{n - 2} & : \text{otherwise} \end{cases}$ $a_n$ has a closed-form expression: :$a_n = \dfrac {3^n - \left({-2}\right)^n} 5$


Recursively Defined Sequence/Examples/Minimum over k of Maximum of 1 plus Function of k and 2 plus Function of n-k

Consider the integer sequence $\left\langle{f \left({n}\right)}\right\rangle$ defined recusrively as: :$f \left({n}\right) = \begin{cases} 0 & : n = 1 \\ \displaystyle \min_{0 \mathop < k \mathop < n} \max \left({1 + f \left({k}\right), 2 + f \left({n - k}\right)}\right) & : n > 1 \end{cases}$ $f \left({n}\right)$ has a closed-form expression: :$f \left({n}\right) = m$ for $F_m < n \le F_{m + 1}$ where $F_m$ denotes the $m$th Fibonacci number.


Zeckendorf Representation of Integer shifted Left

Let $\map f x$ be the real function defined as: :$\forall x \in \R: \map f x = \floor {x + \phi^{-1} }$ where: :$\floor {\, \cdot \,}$ denotes the floor function :$\phi$ denotes the golden mean. Let $n \in \Z_{\ge 0}$ be a positive integer. Let $n$ be expressed in Zeckendorf representation: :$n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$ with the appropriate restrictions on $k_1, k_2, \ldots, k_r$. Then: :$F_{k_1 + 1} + F_{k_2 + 1} + \cdots + F_{k_r + 1} = \map f {\phi n}$


Two Non-Negative Integers have Zeckendorf Representations of which one is Shifted Representation of the Other

Let $m, n \in \Z_{\ge 0}$ be non-negative integers. Then there exists a unique set of integers: :$\left\{ {k_1, k_2, \ldots, k_r}\right\}$ where: : $k_1 \gg k_2 \gg \cdots \gg k_r$ where $a \gg b$ denotes that $a - b > 1$ such that: :$m = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$ and: :$n = F_{k_1 + 1} + F_{k_2 + 1} + \cdots + F_{k_r + 1}$ Note that: : each of the $k$'s may be negative and: :$r$ may equal $0$.


Generating Function by Power of Parameter

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Let $m \in \Z_{\ge 0}$ be a non-negative integer. Then $z^m  G \left({z}\right)$ is the generating function for the sequence $\left\langle{a_{n - m} }\right\rangle$.


Generating Function Divided by Power of Parameter

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Let $m \in \Z_{\ge 0}$ be a non-negative integer. Then $\dfrac 1 {z^m} \left({G \left({z}\right) - \displaystyle \sum_{k \mathop = 0}^{m - 1} a_k z^k}\right)$ is the generating function for the sequence $\left\langle{a_{n + m} }\right\rangle$.


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}} {{qed}}


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 $\overline z = -z$ {{iff}} $z$ 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 {{iff}}: :$\dfrac {z_1 - z_3} {z_3 - z_2} = \lambda$ 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:


Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm implies Limit of Sequence


Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential


Equivalence of Definitions of Real Exponential Function/Extension of Rational Exponential implies Differential Equation


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 {{iff}}: ::$\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 = 0$ :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 {{iff}} 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

:$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$


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 $d \ne 0$.


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 {{iff}} 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 $\R$.


Comparison Test for Convergence of Power Series

Let $A = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$ and $B = \displaystyle \sum_{n \mathop \ge 0} b_n z^n$ be power series in $\C$. Let $R_A$ and $R_B$ be the radii of convergence of $A$ and $B$ respectively. Let $\cmod {b_n} \le \cmod {a_n}$ for all $n \in \N$. Then $R_A \le R_B$.


Hyperbolic Tangent of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:


Hyperbolic Cosine of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\cosh \paren {a + b i} = \cosh a \cos b + i \sinh a \sin b$ where: :$\cos$ denotes the real cosine function :$\sin$ denotes the real sine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function


Hyperbolic Tangent of Complex Number/Formulation 1

:$\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$


Hyperbolic Tangent of Complex Number/Formulation 2

:$\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$


Hyperbolic Tangent of Complex Number/Formulation 3

:$\tanh \paren {a + b i} = \dfrac {\tanh a - \tanh a \tan^2 b} {1 + \tan^2 a \tanh^2 b} + \dfrac {\tan b + \tanh^2 a \tan b} {1 + \tanh^2 a \tan^2 b} i$


Hyperbolic Cosecant of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\map \csch {a + b i} = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b + \cosh^2 a \sin^2 b}$ where: :$\csch$ denotes the hyperbolic cosecant function. :$\sin$ denotes the real sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function


Hyperbolic Secant of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\sech \paren {a + b i} = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}$ where: :$\sech$ denotes the hyperbolic secant function. :$\sin$ denotes the real sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function


Hyperbolic Cotangent of Complex Number


Hyperbolic Cotangent of Complex Number/Formulation 2

:$\map \coth {a + b i} = \dfrac {1 + i \coth a \cot b} {\coth a - i \cot b}$


Hyperbolic Cotangent of Complex Number/Formulation 3

:$\map \coth {a + b i} = \dfrac {\coth a - \coth a \cot^2 b} {\coth^2 a + \cot^2 b} + \dfrac {\coth b + \coth^2 a \cot b} {\coth^2 a + \cot^2 b} i$


Hyperbolic Tangent of Complex Number/Formulation 4

:$\tanh \paren {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$


Tangent of Complex Number/Formulation 4

:$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$