Fourier Series/Square of x minus pi, Square of pi

:400pxrightthumb$\map f x$ and $5$th order expansion Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as: :$\map f x = \begin{cases} \paren {x - \pi}^2 & : 0 < x \le \pi \\ \pi^2 & : \pi < x < 2 \pi \end{cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {2 \pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\frac {2 \cos n x} {n^2} + \paren {\frac {\paren {-1}^n \pi} n - \frac {2 \paren {1 - \paren {-1}^n} } {\pi n^3} } \sin n x}       | c =  }} {{eqn | r = \frac {2 \pi^2} 3 + 2 \paren {\cos x + \frac {\cos 2 x} {2^2} + \frac {\cos 3 x} {3^2} + \cdots}       | c =  }} {{eqn | o =        | ro= -       | r = \pi \sin x + \frac \pi 2 \sin x - \frac \pi 3 \sin x + \frac \pi 4 \sin x - \cdots       | c =  }} {{eqn | o =        | ro= -       | r = \dfrac 4 \pi \sin x - \frac 4 {\pi 3^3} \sin 3 x - \frac 4 {\pi 5^3} \sin 5 x \cdots       | c =  }} {{end-eqn}}


Sum of Reciprocals of Squares Alternating in Sign

{{begin-eqn}} {{eqn | l = \dfrac {\pi^2} {12}       | r = \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^2}       | c =  }} {{eqn | r = \frac 1 {1^2} - \frac 1 {2^2} + \frac 1 {3^2} - \frac 1 {4^2} + \cdots       | c =  }} {{end-eqn}}


Fourier Series/4 minus x squared over Range of 2

Let $\map f x$ be the real function defined on $\openint 0 2$ as: :600pxthumbright$\map f x$ and its $7$th approximation :$\map f x = 4 - x^2$ Then its Fourier series can be expressed as: :$\map f x \sim \displaystyle \frac 8 3 - \frac 4 {\pi^2} \sum_{n \mathop = 1}^\infty \frac {\cos n \pi x} {n^2} + \frac 4 \pi \sum_{n \mathop = 1}^\infty  \frac {\sin n \pi x} n$


Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1

Let $\map f x$ be the real function defined on $\openint {-1} 1$ as: :800pxthumbright$\map f x$ and its $7$th approximation :$\map f x = \begin{cases} 1 & : -1 < x < 0 \\ \map \cos {\pi x} & : 0 < x < 1 \end{cases}$ Then its Fourier series can be expressed as: :$\map f x \sim \displaystyle \dfrac 1 2 + \frac {\cos \pi x} 2 + \sum_{r \mathop = 1}^\infty \paren {\dfrac {4 r \sin 2 r \pi x} {\pi \paren {2 r + 1} \paren {2 r - 1} } - \dfrac {2 \map \sin {2 r + 1} \pi x } {\pi \paren {2 r + 1} } }$


Dilogarithm of Square

:$\operatorname {Li_2} \left({z}\right) + \operatorname {Li_2} \left({-z}\right) = \frac 1 2 \operatorname {Li_2} \left({z^2}\right)$ where $\operatorname {Li_2}$ denotes Spence's Function.


Power Series Expansion for Spence's Function

Spence's function has a power series expansion:  :$\displaystyle \operatorname {Li}_2 \left({z}\right) = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$ This converges for $\left|{z}\right| \le 1$. {{explain|The domain needs to be clarified. As it stands, the notation and construction of the proof suggests $\R$.}}


Odd Function of Zero is Zero

Let $f: \R \to \R$ be an odd function. Let $f$ be defined at the point $x = 0$. Then: :$\map f 0 = 0$


Fourier Cosine Coefficients for Even Function over Symmetric Range

Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$. Let the Fourier series of $\map f x$ be expressed as: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ Then for all $n \in \Z_{\ge 0}$: :$a_n = \dfrac 2 \lambda \displaystyle \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$


Fourier Sine Coefficients for Even Function over Symmetric Range

Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$. Let the Fourier series of $\map f x$ be expressed as: :$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ Then for all $n \in \Z_{> 0}$: :$b_n = 0$


Fourier Series for Even Function over Symmetric Range

Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$. Then the Fourier series of $\map f x$ can be expressed as: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$ where for all $n \in \Z_{\ge 0}$: :$a_n = \dfrac 2 \lambda \displaystyle \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$


Fourier Cosine Coefficients for Odd Function over Symmetric Range

Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$. Let the Fourier series of $\map f x$ be expressed as: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ Then for all $n \in \Z_{\ge 0}$: :$a_n = 0$


Fourier Sine Coefficients for Odd Function over Symmetric Range

Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$. Let the Fourier series of $\map f x$ be expressed as: :$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ Then for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$


Fourier Series for Odd Function over Symmetric Range

Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$. Then the Fourier series of $\map f x$ can be expressed as: :$\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$


Half-Range Fourier Sine Series over Negative Range

Let $\map f x$ be a real function defined on the open real interval $\openint 0 \lambda$. Let $f$ be expressed using the half-range Fourier sine series over $\openint 0 \lambda$: :$\displaystyle \map S x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$ for all $n \in \Z_{\ge 0}$. Then over the interval $\openint {-\lambda} 0$, $\map S x$ takes the values: :$\map S x = -\map f {-x}$ {{refactor|This following bit depends upon what happens at $x {{=}} 0$ which needs to be carefully considered, so put it here as a corollary}} That is, the real function expressed by the half-range Fourier sine series over $\openint 0 \lambda$ is an odd function over $\openint {-\lambda} \lambda$.


Half-Range Fourier Cosine Series over Negative Range

Let $\map f x$ be a real function defined on the open real interval $\openint 0 \lambda$. Let $f$ be expressed using the half-range Fourier cosine series over $\openint 0 \lambda$: :$\displaystyle \map C x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$ where: :$a_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$ for all $n \in \Z_{\ge 0}$. Then over the closed real interval $\openint {-\lambda} 0$, $\map C x$ takes the values: :$\map C x = \map f {-x}$ {{refactor|This following bit depends upon what happens at $x {{=}} 0$ which needs to be carefully considered, so put it here as a corollary}} That is, the real function expressed by the half-range Fourier cosine series over $\openint 0 \lambda$ is an even function over $\openint {-\lambda} \lambda$.


Fourier Series/x over 0 to 2, x-2 over 2 to 4

Let $\map f x$ be the real function defined on $\openint 0 4$ as: :$\map f x = \begin{cases} x & : 0 < x \le 2 \\ x - 2 & : 2 < x < 4 \end{cases}$ Then its Fourier series can be expressed as: :$\map f x \sim \displaystyle 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \paren {1 + \frac {4 \paren {-1}^r} {\paren {2 r - 1} \pi} } \cos \frac {\paren {2 r - 1} \pi x} 4$


Fourier Series over General Range from Specific

Let $a, b \in \R$ be real numbers. Let $f: \R \to \R$ be a function such that $\displaystyle \int_a^b \map f x \rd x$ converges absolutely. Then $f$ can be expressed by a Fourier series of the form: :$\displaystyle \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$ where: {{begin-eqn}} {{eqn | l = A_m       | r = \dfrac 2 {b - a} \int_a^b \map f x \cos \frac {2 m \pi \paren {x - a} } {b - a} \rd x }} {{eqn | l = B_m       | r = \dfrac 2 {b - a} \int_a^b \map f x \sin \frac {2 m \pi \paren {x - a} } {b - a} \rd x }} {{end-eqn}}


Definite Integral to Infinity of Reciprocal of 1 plus Power of x

:$\displaystyle \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \csc \paren {\frac \pi n}$


Leibniz's Integral Rule

Let $\map f {x, t}$, $\map a t$, $\map b t$ be continuously differentiable real functions on some region $R$ of the $\tuple {x, t}$ plane. Then for all $\tuple {x, t} \in R$: :$\displaystyle \frac \rd {\rd t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x = \map f {t, \map b t} \frac {\rd b} {\rd t} - \map f {t, \map a t} \frac {\rd a} {\rd t} + \int_{\map a t}^{\map b t} \frac {\partial} {\partial t} \map f {x, t} \rd x$


Series Expansion of Function over Complete Orthonormal Set

Let $\map f x$ be a real function defined over the interval $\openint a b$. Let $\map f x$ be able to be expressed in terms of a complete orthonormal set of real functions $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ for some indexing set $I$: :$\map f x = \displaystyle \sum_{i \mathop \in I} a_i \map {\phi_i} x$ Then the coefficients $\family {a_i}_{i \mathop \in I}$ can be determined as: :$\forall i \in I: a_i = \displaystyle \int_a^b \map f x \map {\phi_i} x \rd x$


Vector as Sum of Orthogonal Base Vectors

Let $\mathbf v$ be a vector quantity in ordinary $3$-space. Let $\mathbf i, \mathbf j, \mathbf k$ be orthonormal base vectors. Then: :$\mathbf v = \paren {\mathbf v \cdot \mathbf i} \mathbf i + \paren {\mathbf v \cdot \mathbf j} \mathbf j + \paren {\mathbf v \cdot \mathbf k} \mathbf k$


Scaled Sine Functions of Integer Multiples form Orthonormal Set

For all $n \in \Z_{>0}$, let $\map {\phi_n} x$ be the real function defined on the interval $\openint 0 \lambda$ as: :$\map {\phi_n} x = \sqrt {\dfrac 2 \lambda} \sin \dfrac {n \pi x} \lambda$ Let $S$ be the set: :$S = \set {\phi_n: n \in \Z_{>0} }$ Then $S$ is an orthonormal set.


Correspondence Theorem for Quotient Rings

Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq A$ be an ideal. Let $A / \mathfrak a$ be the quotient ring and $\pi : A \to A / \mathfrak a$ the quotient ring epimorphism.


Correspondence Theorem for Quotient Rings/Bijection

The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between the ideals of $A$ containing $\mathfrak a$ and the ideals of $A/\mathfrak a$, specifically: Let $I$ be the set of ideals of $A$ containing $\mathfrak a$. Let $J$ be the set of ideals of $A/\mathfrak a$. Then: #For every ideal $\mathfrak b \in I$, its image $\pi^{\to}(\mathfrak b) = \pi(\mathfrak b) \in J$. #For every ideal $\mathfrak c \in J$, its preimage $\pi^{\gets}(\mathfrak c) = \pi^{-1}(\mathfrak c) \in I$. #The restrictions $\pi^\to : I \to J$ and $\pi^\gets : J \to I$ are reverse bijections.


Correspondence Theorem for Ring Epimorphisms/Bijection

The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between $I$ and $J$, specifically: #For every ideal $\mathfrak a \in I$, its image $\pi^{\to}(\mathfrak a) = \pi(\mathfrak a) \in J$. #For every ideal $\mathfrak b \in J$, its preimage $\pi^{\gets}(\mathfrak b) = \pi^{-1}(\mathfrak b) \in I$. #The restrictions $\pi^\to : I \to J$ and $\pi^\gets : J \to I$ are reverse bijections.


Image of Ideal under Ring Epimorphism is Ideal

Let $f : A \to B$ be a ring epimorphism. Let $I \subseteq A$ be an ideal. Then its image $f(I) \subseteq B$ is an ideal.


Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal

Let $A$ and $B$ be commutative rings with unity. Let $f : A \to B$ be a ring homomorphism. Let $\mathfrak p \subseteq B$ be a prime ideal. Then its preimage $\map {f^{-1} } {\mathfrak p}$ is a prime ideal of $A$.


Correspondence Theorem for Localizations of Rings

Let $A$ be a commutative ring with unity. Let $S\subseteq A$ be a multiplicatively closed subset. Let $A \overset \iota \to A_S$ be the localization at $S$. Let $I$ be the set of saturated ideals of $A$ by $S$. Let $J$ be the set of ideals of $A_S$.


Integral Domain iff Zero Ideal is Prime

Let $A$ be a commutative ring with unity. {{TFAE}} :$(1): \quad A$ is an integral domain :$(2): \quad$ the zero ideal $0 \subseteq A$ is prime


Complement of Prime Ideal of Ring is Multiplicatively Closed

Let $R$ be a commutative ring with unity. Let $P \subset R$ be a prime ideal of $R$. Then its complement $R \setminus P$ is multiplicatively closed.


Derivative of Gamma Function

:$\displaystyle \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$ where $\map {\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$.


Hankel Representation of Riemann Zeta Function

Let $C$ be the Hankel contour. Then for $s \in \C \setminus \Z_{>0}$:  :$\displaystyle \zeta\left({s}\right) = \frac {i \Gamma \left({1 - s}\right)} {2 \pi} \oint_C \frac {\left({-z}\right)^{s - 1} } {e^z - 1} \, \mathrm d z$ where: :$\zeta$ is the Riemann Zeta function :$\Gamma$ is the Gamma function.


Fourier Series/Minus Pi over 0 to Pi, x minus Pi over Pi to 2 Pi

Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as: :500pxthumbright$\map f x$ and its $7$th approximation :$\map f x = \begin{cases} -\pi & : 0 < x \le \pi \\ x - \pi & : \pi < x < 2 \pi \end{cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = -\frac \pi 4 + \frac 2 \pi \sum_{r \mathop = 0}^\infty \frac {\cos \paren {2 r + 1} x} {\paren {2 r + 1}^2} - \sum_{n \mathop = 1}^\infty \frac {2 - \paren {-1}^n \sin n x} n       | c =  }} {{eqn | r = -\frac \pi 4 + \dfrac 2 \pi \paren {\cos x + \frac {\cos 3 x} 9 + \frac {\cos 5 x} {25} + \cdots} - 3 \paren {\sin x + \frac {\sin 3 x} 3 + \frac {\sin 5 x} 5 + \cdots} - \paren {\frac {\sin 2 x} 2 + \frac {\sin 4 x} 4 + \cdots}       | c =  }} {{end-eqn}}


Argument Principle

Let $f$ be a function meromorphic in the interior of some simply connected region $D$. Let $f$ be holomorphic with no zeroes on the boundary of $D$. Let $N$ denote the number of zeroes of $f$ in the interior of $D$, counted up to multiplicity. Let $P$ denote the number of poles of $f$ in the interior of $D$, counted up to order. Then:   :$\displaystyle N - P = \frac 1 {2\pi i} \oint_D \frac { f'\left({z}\right) } { f\left({z}\right) } \, \mathrm d z$


Ring with Unity has Prime Ideal

Let $A$ be a non-trivial commutative ring with unity. Then $A$ has a prime ideal.


Proper Ideal of Ring is Contained in Maximal Ideal

Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq A$ be a proper ideal. Then there exists a maximal ideal $\mathfrak m$ with $\mathfrak a \subseteq \mathfrak m$.


Proper Ideal iff Quotient Ring is Nontrivial

Let $A$ be a commutative ring. Let $\mathfrak a \subseteq A$ be an ideal. {{TFAE}} :$(1): \quad \mathfrak a$ is a proper ideal :$(2): \quad$ The quotient ring $A / \mathfrak a$ is nontrivial ring


Half-Range Fourier Sine Series/Sine of Half x over 0 to Pi, Minus Sine of Half x over Pi to 2 Pi

Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as: :600pxthumbright$\map f x$ and its $7$th approximation :$\map f x = \begin {cases} \sin \dfrac x 2 & : 0 \le x < \pi \\ -\sin \dfrac x 2 & : \pi < x \le 2 \pi \end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac 8 \pi \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {n \sin n x} {4 n^2 - 1}       | c =  }} {{eqn | r = \frac 8 \pi \paren {\frac {\sin x} {1 \times 3} - \frac {2 \sin 2 x} {3 \times 5} + \frac {3 \sin 3 x} {5 \times 7} - \frac {4 \sin 4 x} {7 \times 9} + \frac {5 \sin 5 x} {9 \times 11} - \dotsb}       | c =  }} {{end-eqn}}


Fourier Series/Cosine of x over Minus Pi to Zero, Minus Cosine of x over Zero to Pi

Let $\map f x$ be the real function defined on $\openint {-\pi} \pi$ as: :600pxthumbright$\map f x$ and its $7$th approximation :$\map f x = \begin {cases} \cos x & : -\pi < x < 0 \\ -\cos x & : 0 < x < \pi \end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = -\frac 8 \pi \sum_{r \mathop = 1}^\infty \frac {r \sin 2 r x} {4 r^2 - 1}       | c =  }} {{eqn | r = -\frac 8 \pi \paren {\frac {\sin 2 x} {1 \times 3} + \frac {\sin 4 x} {3 \times 5} + \frac {\sin 6 x} {5 \times 7}  + \frac {\sin 8 x} {7 \times 9} + \dotsb}       | c =  }} {{end-eqn}}


Holomorphic Function is Analytic

Let $a \in \C$ be a complex number. Let $r > 0$ be a real number. Let $f$ be a function holomorphic on some open ball $D = \map B {a, r}$.  Then $f$ is complex analytic on $D$.


Schwarz's Lemma

Let $D$ be the unit disk centred at $0$. Let $f : D \to \C$ be a holomorphic function. Let $\map f 0 = 0$ and $\cmod {\map f z} \le 1$ for all $z \in D$.  Then $\cmod {\map {f'} 0} \le 1$, and $\cmod {\map f z} \le \cmod z$ for all $z \in D$.


Ring of Integers of Number Field is Dedekind Domain

Let $K$ be a number field. Let $\mathcal O_K$ be its ring of integers. Then $\mathcal O_K$ is a Dedekind domain.


Fourier Series/Exponential of x over Minus Pi to Pi

Let $\map f x$ be the real function defined on $\R$ as: :200pxthumbright$\map f x$ and its $7$th approximation :$\map f x = \begin{cases} e^x & : -\pi < x \le \pi \\ \map f {x + 2 \pi} & : \text{everywhere} \end{cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {\sinh \pi} \pi \paren {1 + 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {1 + n^2} \paren {\cos n x - n \sin n x} }       | c =  }} {{eqn | r = \frac {\sinh \pi} \pi \paren {1 + 2 \paren {-\dfrac {\cos x - \sin x} 2 + \dfrac {\cos 2 x - 2 \sin 2 x} 5 - \dfrac {\cos 3 x - 3 \sin 3 x} {10} + \dotsb} }       | c =  }} {{end-eqn}}


Sum of Reciprocals of Squares plus 1

:$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^2 + 1}  = \frac 1 2 \paren {\pi \coth \pi - 1}$


Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi

Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer. :500pxthumbright$\map f x$ for $\lambda = 4 \cdotp 6$ and its $5$th approximation Let $\map f x$ be the real function defined on $\openint 0 \pi$ as: :$\map f x = \cos \lambda x$ Then its half-range Fourier cosine series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} {\lambda^2 - n^2} }       | c =  }} {{eqn | r = \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} - \frac {\cos x} {\lambda^2 - 1} + \frac {\cos 2 x} {\lambda^2 - 4} - \frac {\cos 3 x} {\lambda^2 - 9} + \frac {\cos 4 x} {\lambda^2 - 16} - \dotsb}       | c =  }} {{end-eqn}}


Series Expansion for Pi Cotangent of Pi Lambda

Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer. Then: :$\displaystyle \pi \cot \pi \lambda = \frac 1 \lambda + \sum_{n \mathop = 1}^\infty \frac {2 \lambda} {\lambda^2 - n^2}$


Series Expansion for Pi Cosecant of Pi Lambda

Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer. Then: :$\displaystyle \pi \csc \pi \lambda = \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac 1 {n + \lambda} + \frac 1 {n - 1 - \lambda} }$


Prime Ideal of Principal Ideal Domain is Maximal

Let $D$ be a principal ideal domain whose zero is $0_D$. Let $J \subseteq D$ be a nonzero prime ideal. Then $J$ is maximal.


Gauss's Digamma Theorem

Let $\dfrac p q$ be a positive rational number with $p < q$. Then:  :$\displaystyle \psi \left({\frac p q}\right) = -\gamma - \ln 2 q - \frac \pi 2 \cot \left({\frac p q \pi}\right) + 2 \sum_{n \mathop = 1}^{\left\lceil{q / 2}\right\rceil - 1} \cos \left({\frac {2 \pi p n} q}\right) \ln\left({\sin \left({\frac {\pi n} q}\right)}\right)$ where: :$\psi$ is the digamma function :$\cot$ is the cotangent function :$\ln$ is the natural logarithm.


Gauss's Integral Form of Digamma Function

Let $z$ be a complex number with a positive real part, then:  :$\displaystyle \psi \left({z}\right) = \int_0^\infty \left({\frac{ e^{-t} } t - \frac {e^{-zt } } {1 - e^{-t} } }\right) \rd t$ where $\psi$ is the digamma function.


Rodrigues' Formula for Legendre Polynomials

:$\map {P_n} x = \dfrac 1 {2^n n!} \dfrac {\d^n} {\d x^n} \paren {x^2 - 1}^n$ where $P_n$ is the $n$th Legendre polynomial.


Modulus of Gamma Function of Imaginary Number

Let $t \in \R$ be a real number. Then:  :$\cmod {\map \Gamma {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$ where: :$\Gamma$ is the Gamma function :$\csch$ is the hyperbolic cosecant function.


Existence of Homomorphism between Localizations of Ring

Let $A$ be a commutative ring with unity. Let $S, T \subseteq A$ be multiplicatively closed subsets. {{TFAE}} :$(1): \quad$ There exists an $A$-algebra homomorphism $h : A_S \to A_T$ between localizations, the '''induced homomorphism'''. :$(2): \quad S$ is a subset of the saturation of $T$. :$(3): \quad$  The saturation of $S$ is a subset of the saturation of $T$. :$(4): \quad$  Every prime ideal meeting $S$ also meets $T$.


Fourier Series/Pi Squared minus x Squared over Minus Pi to Pi

Let $\map f x$ be the real function defined on $\openint {-\pi} \pi$ as: :400pxthumbright$\map f x$ and its $4$th approximation :$\map f x = \pi^2 - x^2$ $f$ can be expressed as a half-range Fourier cosine series thus: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {2 \pi^2} 3 + 4 \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {\cos n x} {n^2}       | c =  }} {{eqn | r = \frac {2 \pi^2} 3 + 4 \paren {\cos x - \frac 1 4 \cos 2 x + \frac 1 9 \cos 3 x - \cdots}       | c =  }} {{end-eqn}}


Half-Range Fourier Cosine Series/Identity Function/0 to Pi

Let $\map f x$ be the real function defined on $\openint 0 \pi$ as: :400pxthumbright$\map f x$ and its $4$th approximation :$\map f x = x$ Then its half-range Fourier cosine series can be expressed as: {{begin-eqn}} {{eqn | l = x       | o = \sim       | r = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\cos \paren {2 n - 1} x} {\paren {2 n - 1}^2}       | c =  }} {{eqn | r = \frac \pi 2 - \frac 4 \pi \paren {\cos x + \frac {\cos 3 x} {3^2} + \frac {\cos 5 x} {5^2} + \cdots}       | c =  }} {{end-eqn}}


Hilbert's Basis Theorem for Finitely Generated Algebras

Let $A$ be a Noetherian ring. Let $B$ be a finitely generated algebra over $A$. Then $B$ is Noetherian.


Zariski's Lemma

Let $L/k$ be a field extension. Let $L$ be finitely generated as an algebra over $k$. Then $L/k$ is a finite field extension.


Cayley-Hamilton Theorem/Finitely Generated Modules

Let $A$ be a commutative ring with unity. Let $M$ be a finitely generated $A$-module. Let $\mathfrak a$ be an ideal of $A$. Let $\phi$ be an endomorphism of $M$ such that $\phi \left({M}\right) \subseteq \mathfrak a M$. Then $\phi$ satisfies an equation of the form: :$\phi^n + a_{n-1} \phi^{n-1} + \cdots + a_1 \phi + a_0 = 0$ with the $a_i \in \mathfrak a$.


Weak Nullstellensatz

Let $k$ be an algebraically closed field. Let $n \geq 0$ be an natural number. Let $k \left[{x_1,\ldots, x_n}\right]$ be the polynomial ring in $n$ variables over $k$. Let $I \subseteq k \left[{x_1,\ldots, x_n}\right]$ be an ideal. {{TFAE}} # $I$ is the unit ideal: $I = (1)$. # Its zero-locus is empty set: $V(I) = \varnothing$.


Finite-Dimensional Integral Domain over Field is Field

Let $k$ be a field. Let $R$ be a $k$-algebra of finite dimension which is an integral domain. Then $R$ is a field.


Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal

Let $k$ be a field. Let $A$ and $B$ be $k$-algebras. Let $f : A \to B$ be a $k$-algebra homomorphism. Let $B$ be finitely generated over $k$. Let $\mathfrak m$ be a maximal ideal of $B$. Then its preimage $f^{-1}(\mathfrak m)$ is a maximal ideal of $A$.


Subalgebra of Finite Field Extension is Field

Let $E / F$ be an algebraic field extension. Let $A \subseteq E$ be a unital subalgebra over $F$. Then $A$ is a field.


Weierstrass's Elliptic Function is Doubly Periodic

Weierstrass's elliptic function $\map \wp {z; \omega_1, \omega_2}$ is doubly periodic in $2 \omega_1$ and $2 \omega_2$.


Differential Equation satisfied by Weierstrass's Elliptic Function

The differential equation:  :$\paren {\dfrac {\d f} {\d z} }^2 = 4 f^3 - g_2 f - g_3$ where:  :$\displaystyle g_2 = 60 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^4}$ and:  :$\displaystyle g_3 = 140 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^6}$ has the general solution:  :$\map f z = \map \wp {z + C; \omega_1, \omega_2}$ where: :$\wp$ is Weierstrass's elliptic function :$C$ is an arbitrary constant :$\omega_1$, $\omega_2$ are constants independent of $z$. {{explain|What domain are all the variables and constants in?}}


Piecewise Continuous Function with Improper Integrals may not be Bounded

Let $f$ be a real function defined on a closed interval $\closedint a b$, $a < b$.  Let $f$ be a piecewise continuous function with improper integrals. Then $f$ may not be piecewise continuous and bounded on $\closedint a b$.


Weierstrass's Elliptic Function is Even in z

Let $\omega_1$ and $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.  For $z \in \C \setminus \left\{2m \omega_1 + 2n\omega_2: \left({n,m}\right) \in \Z^2\right\}$: :$\displaystyle \wp\left({-z; \omega_1, \omega_2}\right) = \wp\left({z; \omega_1, \omega_2}\right)$ That is, Weierstrass's elliptic function is even in $z$.


Duplication Formula for Weierstrass's Elliptic Function

Let $\omega_1$, $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part. Let $z$ be a complex number where $z \notin \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$. Then: :$\map \wp {2 z; \omega_1, \omega_2} = \dfrac 1 4 \paren {\dfrac {\map {\wp''} {z; \omega_1, \omega_2} } {\map {\wp'} {z; \omega_1, \omega_2} } }^2 - 2 \map \wp {z; \omega_1, \omega_2}$ where: :$\wp$ is Weierstrass's elliptic function  :$\wp'$ and $\wp''$ denote its first and second derivative with respect to $z$.


Integral to Infinity of Sine p x over x

:$\displaystyle \int_0^\infty \frac {\sin p x} x \rd x = \begin{cases} \dfrac \pi 2 & : p > 0 \\ 0 & : p = 0 \\ -\dfrac \pi 2 & : p < 0 \end{cases}$


Radical of Sum of Ideals

Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals. Then for the radical of their sum we have: :$\operatorname{Rad} \left({\mathfrak a + \mathfrak b}\right) = \operatorname{Rad} \left({\operatorname{Rad} \left({\mathfrak a}\right) + \operatorname{Rad} \left({\mathfrak b}\right)}\right)$


Ideal of Ring is Contained in Radical

Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq A$ be an ideal. Then $\mathfrak a$ is contained in its radical: :$\mathfrak a \subseteq \operatorname{Rad} \left({\mathfrak a}\right)$


Radical of Ideal Preserves Inclusion

Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq \mathfrak b \subseteq A$ be an ideals. Then we have an inclusion of their radicals: :$\operatorname{Rad} \left({\mathfrak a}\right) \subseteq \operatorname{Rad} \left({\mathfrak b}\right)$


Ideals with Coprime Radicals are Coprime

Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals. Let their radicals be coprime: :$\operatorname{Rad} \left({\mathfrak a}\right) + \operatorname{Rad} \left({\mathfrak b}\right) = \left({1}\right)$ Then $\mathfrak a$ and $\mathfrak b$ are coprime.


Radical of Unit Ideal

Let $A$ be a commutative ring with unity. Let $\left({1}\right)$ be its unit ideal. Then its radical equals $(1)$: :$\operatorname{Rad} \left({\left({1}\right)}\right) = \left({1}\right)$.


Radical of Power of Prime Ideal

Let $A$ be a commutative ring with unity. Let $\mathfrak p \subseteq A$ be a prime ideal. Let $n > 0$ be a natural number. Then the radical of the $n$th power of $\mathfrak p$ equals $\mathfrak p$: :$\operatorname{Rad} \left({\mathfrak p^n}\right) = \mathfrak p$.


Coefficients of Product of Two Polynomials

Let $R$ be a commutative ring with unity. Let $f, g \in R \sqbrk x$ be polynomials over $R$. For a natural number $n \ge 0$, let: :$a_n$ be the coefficient of the monomial $x^n$ in $f$. :$b_n$ be the coefficient of the monomial $x^n$ in $g$.


Equivalence of Definitions of Irreducible Polynomial over Field

Let $K$ be a field. {{TFAE|def = Irreducible Polynomial}}


Linear Combination of Integrals/Indefinite

:$\displaystyle \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$


Linear Combination of Integrals/Definite

:$\displaystyle \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$


Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories

Let $\Z$ be the ring of integers. Let $\mathbf{Ab}$ be the category of abelian groups. Let $\mathbf{\mathbb Z-Mod}$ be the category of unitary $\Z$-modules. Then the: * forgetful functor $\mathbf{\mathbb Z-Mod} \to \mathbf{Ab}$ * associated Z-module functor $\mathbf{Ab} \to \mathbf{\mathbb Z-Mod}$ are strict inverse functors. In particular, $\mathbf{Ab}$ and $\mathbf{\mathbb Z-Mod}$ are isomorphic.


Correspondence between Abelian Groups and Z-Modules/Homomorphisms

Let $G, H$ be abelian groups. Let $f : G \to H$ be a mapping. {{TFAE}} # $f$ is a group homomorphism. # $f$ is a $\Z$-module homomorphism between the $\Z$-modules associated with $G$ and $H$.


Mapping to Singleton is Unique

Let $S$ be a set. Let $T$ be a singleton. Then there exists a unique mapping $S \to T$.


Rectangular Formula for Definite Integrals

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$: :$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$ Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\displaystyle \int_a^b \map f x \rd x \approx \sum_{r \mathop = 0}^{n - 1} h \map f {x_r}$ where $h = \dfrac {b - a} n$.


Trapezoidal Formula for Definite Integrals

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$: :$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$ Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\displaystyle \int_a^b \map f x \rd x \approx \dfrac h 2 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{n - 1} 2 \map f {x_r} }$ where $h = \dfrac {b - a} n$.


Simpson's Rule

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$: :$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$ where $n$ is even. Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\displaystyle \int_a^b \map f x \rd x \approx \dfrac h 3 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{m - 1} 2 \map f {x_{2 m - 1} } + \sum_{r \mathop = 1}^{m - 1} 4 \map f {x_{2 m} } }$ where: :$h = \dfrac {b - a} n$ :$m = \dfrac n 2$


Definite Integral to Infinity of Reciprocal of x Squared plus a Squared

:$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$ for $a \ne 0$.


Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary

:$\displaystyle \int_0^\infty \frac 1 {a^n + x^n} \rd x = \frac \pi {n a^{n - 1} } \csc \paren {\frac \pi n}$


Definite Integral to Infinity of Power of x over 1 + x

:$\displaystyle \int_0^\infty \dfrac {x^{p - 1} \rd x} {1 + x} = \frac \pi {\sin \pi p}$ for $0 < p < 1$.


Universal Property of Free Abelian Group on Set

Let $S$ be a set. Let $(\Z^{(S)}, \iota)$ be the free abelian group on $S$. Let $G$ be an abelian group. Let $f : S \to G$ be a mapping. Then there exists a unique group homomorphism $g : \Z^{(S)} \to G$ with $g \circ \iota = f$: :$\xymatrix{ S \ar[d]_\iota \ar[r]^{\forall f} & G\\ \Z^{(S)} \ar[ru]_{\exists ! g} }$


Definite Integral from 0 to a of Root of a Squared minus x Squared

:$\displaystyle \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$ for $a > 0$.


Universal Property of Abelianization of Group

Let $G$ be a group. Let $G^{\operatorname {ab} }$ be its abelianization. Let $\pi : G \to G^{\operatorname {ab} }$ be the quotient group epimorphism. Let $H$ be an abelian group. Let $f: G \to H$ be a group homomorphism. Then there exists a unique group homomorphism $g : G^{\operatorname {ab}} \to H$ such that $g \circ \pi = f$: :$\xymatrix { G \ar[d]_\pi \ar[r]^{\forall f} & H\\ G^{\operatorname {ab} } \ar[ru]_{\exists ! g} }$


Definite Integral from 0 to a of x^m by (a^n - x^n)^p

:$\displaystyle \int_0^a x^m \paren {a^n - x^n}^p \rd x = \frac {a^{m + 1 + n p} \, \map \Gamma {\frac {m + 1} n} \map \Gamma {p + 1} } {n \map \Gamma {\frac {m + 1} n + p + 1} }$


Equivalence of Definitions of Complement of Subgroup

Let $G$ be a group with identity $e$. Let $H$ and $K$ be subgroups. {{TFAE|def = Complement of Subgroup}}


Equivalent Characterizations of Finer Equivalence Relation

Let $X$ be a set. Let $\equiv$ and $\sim$ be equivalence relations on $X$. {{TFAE}} # $\equiv$ is finer than $\sim$: #:$\forall x, y \in X : x \equiv y \implies x \sim y$ #The graph of $\equiv$ is contained in the graph of $\sim$. #Every $\equiv$-equivalence class is contained in a $\sim$-equivalence class. #Every $\sim$-equivalence class is saturated under $\equiv$.


Equivalence of Definitions of Finer Topology

Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. {{TFAE|def = Finer Topology}}


Definite Integral from 0 to Half Pi of Square of Sine x

:$\displaystyle \int_0^{\frac \pi 2} \sin^2 x \rd x = \frac \pi 4$


Definite Integral from 0 to Half Pi of Square of Cosine x

:$\displaystyle \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$


Reduction Formula for Definite Integral of Power of Cosine

Let $n \in \Z_{> 0}$ be a positive integer. Let $I_n$ be defined as: :$\displaystyle I_n = \int_0^{\frac \pi 2} \cos^n x \rd x$ Then $\left\langle{I_n}\right\rangle$ is a decreasing sequence of real numbers which satisfies: :$n I_n = \left({n - 1}\right) I_{n - 2}$ Thus: :$I_n = \dfrac {n - 1} n  I_{n - 2}$ is a reduction formula for $I_n$.


Definite Integral from 0 to Half Pi of Even Power of Cosine x

:$\displaystyle \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$


Definite Integral from 0 to Half Pi of Odd Power of Cosine x

:$\displaystyle \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\left({2^n n!}\right)^2} {\left({2 n + 1}\right)!}$


Arctangent of Zero is Zero

:$\arctan 0 = 0$


Riemann P-symbol in terms of Gaussian Hypergeometric Function

Let: :$\displaystyle f\left({z}\right) = \operatorname P \left\{ \begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} \right\}$ where: :$\operatorname P$ is the Riemann P-symbol :$\alpha + \beta + \gamma + \alpha' + \beta' + \gamma' = 1$  :$\alpha - \alpha'$ is not a negative integer.  Then:  :$\displaystyle f\left({z}\right) = \left({ \frac {z - a} {z - b} }\right)^\alpha \left({ \frac {z - c} {z - b} }\right)^\gamma {}_2 \operatorname F_1 \left({ {\alpha + \beta + \gamma, \alpha + \beta' + \gamma} \atop {1 + \alpha - \alpha'} } \, \middle \vert {\, \frac {\left({z - a}\right) \left({c - b}\right)} {\left({z - b} \right) \left({c - a}\right)} }\right)$ where ${}_2 \operatorname F_1$ is the Gaussian hypergeometric function.


Sum of Powers of Positive Integers

Let $n, p \in \Z_{>0}$ be (strictly) positive integers. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n k^p       | r = 1^p + 2^p + \cdots + n^p       | c =  }} {{eqn | r = \frac {n^{p + 1} } {p + 1} + \sum_{k \mathop = 1}^p \frac {B_k \, p^{\underline {k - 1} } \, n^{p - k + 1} } {k!}       | c =  }} {{eqn | r = \frac {n^{p + 1} } {p + 1} + \frac {B_1 \, n^p} {1!} + \frac {B_2 \, p \, n^{p - 1} } {2!} + \frac {B_4 \, p \paren {p - 1} \paren {p - 2} n^{p - 3} } {4!} + \cdots       | c =  }} {{end-eqn}} where: :$B_k$ are the Bernoulli numbers :$p^{\underline k}$ is the $k$th falling factorial of $p$.


Order of Finite p-Group is Power of p

Let $G$ be a finite group. Let $p$ be a prime number. Let all elements of $G$ have order a power of $p$. Then $G$ is a $p$-group.


Sum of Reciprocals of Sixth Powers Alternating in Sign

{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^6}       | r = \frac 1 {1^6} - \frac 1 {2^6} + \frac 1 {3^6} - \frac 1 {4^6} + \cdots       | c =  }} {{eqn | r = \frac {31 \pi^6} {30 \, 240}       | c =  }} {{end-eqn}}


Sum of Reciprocals of Fourth Powers of Odd Integers

{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\left({2 n - 1}\right)^4}       | r = 1 + \dfrac 1 {3^4} + \dfrac 1 {5^4} + \dfrac 1 {7^4} + \dfrac 1 {9^4} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^4} {96}       | c =  }} {{end-eqn}}


Sum of Reciprocals of Sixth Powers of Odd Integers

{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\left({2 n - 1}\right)^6}       | r = 1 + \dfrac 1 {3^6} + \dfrac 1 {5^6} + \dfrac 1 {7^6} + \dfrac 1 {9^6} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^6} {960}       | c =  }} {{end-eqn}}


Half-Range Fourier Sine Series/x by Pi minus x over 0 to Pi

Let $\map f x$ be the real function defined on $\openint 0 \pi$ as: :$\map f x = x \paren {\pi - x}$ Then its half-range Fourier sine series can be expressed as: :$\displaystyle \map f x \sim \frac 8 \pi \sum_{r \mathop = 0}^\infty \frac {\sin \paren {2 r + 1} x} {\paren {2 r + 1}^3}$


Sum of Reciprocals of Powers of Odd Integers Alternating in Sign

:$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x$ where: :$\map \Re s > 0$ :$\Gamma$ is the gamma function :$\sech$ is the hyperbolic secant function.


Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign in Pairs

{{begin-eqn}} {{eqn | l = \frac 1 {1^3} + \frac 1 {3^3} - \frac 1 {5^3} - \frac 1 {7^3} + \cdots       | r = \frac {3 \pi^3 \sqrt 2} {128}       | c =  }} {{end-eqn}}


Sum of Sequence of Products of Consecutive Odd Reciprocals

{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^n \frac 1 {\paren {2 j + 1} \paren {2 j + 3} }       | r = \frac 1 {1 \times 3} + \frac 1 {3 \times 5} + \frac 1 {5 \times 7} + \frac 1 {7 \times 9} + \cdots + \frac 1 {\paren {2 n + 1} \paren {2 n + 3} }       | c =  }} {{eqn | r = \frac {n + 1} {2 n + 3}       | c =  }} {{end-eqn}}


Equivalence of Definitions of Local Ring Homomorphism

Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ be commutative local rings. Let $f : A \to B$ be a unital ring homomorphism. {{TFAE|def = Local Ring Homomorphism}}


Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals

{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n \frac 1 {j \left({j + 2}\right)}       | r = \frac 1 {1 \times 3} + \frac 1 {2 \times 4} + \frac 1 {3 \times 5} + \frac 1 {4 \times 6} + \cdots + \frac 1 {n \left({n + 2}\right)}       | c =  }} {{eqn | r = \frac 3 4 - \frac {2 n + 3} {2 \left({n + 1}\right) \left({n + 2}\right)}       | c =  }} {{end-eqn}}


Sum of Sequence of Products of Squares of 3 Consecutive Reciprocals

{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2}       | r = \frac 1 {1^2 \times 2^2 \times 3^2} + \frac 1 {2^2 \times 3^2 \times 4^2} + \frac 1 {3^2 \times 4^2 \times 5^2} + \frac 1 {4^2 \times 5^2 \times 6^2} + \cdots       | c =  }} {{eqn | r = \frac {4 \pi^2 - 39} {16}       | c =  }} {{end-eqn}}


Definite Integral from 0 to 1 of Power of u over 1 + Power of u

{{begin-eqn}} {{eqn | l = \int_0^1 \dfrac {u^{a - 1} \rd u} {1 + u^d}       | r = \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {a + j d}       | c =  }} {{eqn | r = \frac 1 a - \frac 1 {a + d} + \frac 1 {a + 2 d} - \frac 1 {a + 3 d} + \cdots       | c =  }} {{end-eqn}} where $a, d > 0$.


Sum of Reciprocals of Even Powers of Odd Integers

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }       | r = \dfrac 1 {1^{2 n} } + \dfrac 1 {3^{2 n} } + \dfrac 1 {5^{2 n} } + \dfrac 1 {7^{2 n} } + \cdots       | c =  }} {{eqn | r = \paren {-1}^{n + 1} \dfrac {B_{2 n} \paren {2^{2 n} - 1} \pi^{2 n} } {2 \paren {2 n}!}       | c =  }} {{end-eqn}}


Sum of Reciprocals of Powers of Integers Alternating in Sign

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^\infty \left({-1}\right)^{j + 1} \frac 1 {j^{2 n} }       | r = \dfrac 1 {1^{2 n} } - \dfrac 1 {2^{2 n} } + \dfrac 1 {3^{2 n} } - \dfrac 1 {4^{2 n} } + \cdots       | c =  }} {{eqn | r = \left({-1}\right)^{n + 1} \dfrac {B_{2 n} \left({2^{2 n - 1} - 1}\right) \pi^{2 n} } {\left({2 n}\right)!}       | c =  }} {{end-eqn}}


Hermite's Formula for Hurwitz Zeta Function

:$\displaystyle \map \zeta {s, q} = \frac 1 {2 q^s} + \frac { q^{1 - s} } {s - 1} + 2 \int_0^\infty \frac {\map \sin {s \arctan \frac x q} } {\paren {q^2 + x^2}^{\frac 1 2 s} \paren {e^{2 \pi x} - 1} } \rd x$ where:  :$\zeta$ is the Hurwitz zeta function :$\map \Re s > 1$ :$\map \Re q > 0$.


Binet's Formula for Logarithm of Gamma Function/Formulation 1

Let $z$ be a complex number with a positive real part. Then:  :$\displaystyle \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + \int_0^\infty \paren {\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} } \frac {e^{-t z} } t \rd t$ where:  :$\Gamma$ is the Gamma function :$\Ln$ is the principal branch of the complex logarithm.


Binet's Formula for Logarithm of Gamma Function/Formulation 2

Let $z$ be a complex number with a positive real part. Then:  :$\displaystyle \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$ where:  :$\Gamma$ is the Gamma function :$\Ln$ is the principal branch of the complex logarithm.


Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign

Let $n \in \Z_{\ge 0}$ be a positive integer. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^\infty \frac {\left({-1}\right)^j} {\left({2 j + 1}\right)^{2 n + 1} }       | r = \left({-1}\right)^{n + 1} \frac {\pi^{2 n + 1} E_{2 n} } {2^{2 n + 2} \left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots       | c =  }} {{end-eqn}} where $E_n$ is the $n$th Euler number.


Sum of Infinite Series of Product of Power and Cosine

Let $r \in \R$ such that $\size r < 1$. Let $z \in \R$ such that $z \ne 2 m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k x       | r = 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots       | c =  }} {{eqn| r = \dfrac {1 - r \cos x} {1 - 2 r \cos x + r^2}       | c =  }} {{end-eqn}}


Sum of Series of Product of Power and Sine

Let $r \in \R$. Let $x \in \R$ such that $x \ne 2 m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n r^k \map \sin {k x}       | r = r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots + r^n \sin n x       | c =  }} {{eqn | r = \dfrac {r \sin x - r^{n + 1} \map \sin {n + 1} x + r^{n + 2} \sin n x} {1 - 2 r \cos x + r^2}       | c =  }} {{end-eqn}}


Euler-Maclaurin Summation Formula

Let $f$ be a real function which is appropriately differentiable and integrable. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map f k       | r = \int_0^n \map f x \rd x - \frac {\map f 0 + \map f n} 2 + \sum_{k \mathop = 1}^\infty \frac {B_{2 k} } {\paren {2 k}!} \paren {\map {f^{\paren {2 k - 1} } } n - \map {f^{\paren {2 k - 1} } } 0}       | c =  }} {{eqn | r = \int_0^n \map f x \rd x - \frac 1 2 \paren {\map f n + \map f 0}       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {12} \paren {\map {f'} n - \map {f'} 0}       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 {720} \paren {\map {f'''} n - \map {f'''} 0}       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {30 \, 240} \paren {\map {f^{\paren 5} } n - \map {f^{\paren 5} } 0}       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 {1 \, 209 \, 600} \paren {\map {f^{\paren 7} } n - \map {f^{\paren 7} } 0}       | c =  }} {{end-eqn}} where: :$f^{\paren k}$ denotes the $k$th derivative of $f$ :$B_n$ denotes the $n$th Bernoulli number. {{Proofread|Confusion between the various types of Bernoulli numbers means I may have got signs and indices wrong.}}


Power Series Expansion of Reciprocal of 1 + x

Let $x \in \R$ such that $-1 < x < 1$. Then: {{begin-eqn}} {{eqn | l = \dfrac 1 {1 + x}       | r = \sum_{k \mathop = 0}^\infty \left({-1}\right)^k x^k       | c =  }} {{eqn | r = 1 - x + x^2 - x^3 + x^4 - \cdots       | c =  }} {{end-eqn}}


Frullani's Integral

:$\displaystyle \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$


Power Series Expansion of Reciprocal of Cube of 1 + x

Let $x \in \R$ such that $-1 < x < 1$. Then: {{begin-eqn}} {{eqn | l = \dfrac 1 {\paren {1 + x}^3}       | r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k       | c =  }} {{eqn | r = 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \cdots       | c =  }} {{end-eqn}}


Power Series Expansion of Square Root of 1 + x

Let $x \in \R$ such that $-1 < x \le 1$. Then: {{begin-eqn}} {{eqn | l = \dfrac 1 {\sqrt {1 + x} }       | r = 1 + \sum_{k \mathop = 1}^\infty \left({-1}\right)^{k - 1} \frac {\left({2 \left({k - 1}\right)}\right)!} {2^{2 k - 1} k! \left({k - 1}\right)!} x^k       | c =  }} {{eqn | r = 1 + \frac 1 2 x - \frac 2 {2 \times 4} x^2 + \frac {1 \times 3} {2 \times 4 \times 6} x^3 - \cdots       | c =  }} {{end-eqn}}


Power Series Expansion for Half Logarithm of 1 + x over 1 - x

{{begin-eqn}} {{eqn | l = \frac 1 2 \map \ln {\frac {1 + x} {1 - x} }       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1} }} {{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots }} {{end-eqn}} valid for all $x \in \R$ such that $-1 < x < 1$.


Power Series Expansion for Logarithm of x/Formulation 1

{{begin-eqn}} {{eqn | l = \ln x        | r = 2 \paren {\sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \paren {\frac {x - 1} {x + 1} }^{2 n + 1} } }} {{eqn | r = 2 \paren {\frac {x - 1} {x + 1} + \frac 1 3 \paren {\frac {x - 1} {x + 1} }^3 + \frac 1 5 \paren {\frac {x - 1} {x + 1} }^5 + \cdots} }} {{end-eqn}} valid for all $x \in \R$ such that $-1 < x < 1$.


Power Series Expansion for Cotangent Function

The (real) cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \cot x       | r = \sum_{n \mathop = 0}^\infty \frac {\left({-1}\right)^{n} 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $0 < \left|{x}\right| < \pi$.


Power Series Expansion for Secant Function

The (real) secant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \sec x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {E_{2 n} x^{2 n} } {\paren {2 n}!} }} {{eqn | r = 1 + \frac {x^2} 2 + \frac {5 x^4} {24} + \frac {61 x^6} {720} + \cdots }} {{end-eqn}} where $E_{2 n}$ denotes the Euler numbers. This converges for $\size x < \dfrac \pi 2$.


Power Series Expansion for Real Arccotangent Function

The arccotangent function has a Taylor series expansion: :$\operatorname {arccot} x = \begin{cases} \displaystyle \frac \pi 2 - \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \displaystyle \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {\left({2 n + 1}\right) x^{2 n + 1} } & : x \ge 1 \\ \displaystyle \pi + \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {\left({2 n + 1}\right) x^{2 n + 1} } & : x \le -1 \end{cases}$ That is: :$\operatorname {arccot} x = \begin{cases} \displaystyle \frac \pi 2 - \left({x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots}\right) & : -1 \le x \le 1 \\ \displaystyle \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \ge 1 \\ \displaystyle \pi + \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \le -1 \end{cases}$


Power Series Expansion for Real Arccosecant Function

The arccosecant function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \operatorname {arccsc} x       | r = \sum_{n \mathop = 0}^\infty \frac {\left({2 n}\right)!} {2^{2 n} \left({n!}\right)^2 \left({2 n + 1}\right) x^{2 n + 1} }       | c =  }} {{eqn | r = \frac 1 x + \frac 1 2 \frac 1 {3 x^3} + \frac {1 \times 3} {2 \times 4} \frac 1 {5 x^5} + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac 1 {7 x^7} + \cdots       | c =  }} {{end-eqn}} which converges for $\left\lvert{x}\right\rvert \ge 1$.


Power Series Expansion for Hyperbolic Sine Function

The hyperbolic sine function has the power series expansion: {{begin-eqn}} {{eqn | l = \sinh x       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.


Power Series Expansion for Hyperbolic Cosine Function

The hyperbolic cosine function has the power series expansion: {{begin-eqn}} {{eqn | l = \cosh x       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.


Power Series Expansion for Hyperbolic Tangent Function

The hyperbolic tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \tanh x       | r = \sum_{n \mathop = 1}^\infty \frac {2^{2 n} \left({2^{2 n} - 1}\right) B_{2 n} \, x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = x - \frac {x^3} 3 + \frac {2 x^5} {15} - \frac {17 x^7} {315} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $\left|{x}\right| < \dfrac \pi 2$.


Power Series Expansion for Hyperbolic Cotangent Function

The hyperbolic cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \coth x       | r = \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 x + \frac x 3 - \frac {x^3} {45} + \frac {2 x^5} {45} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $0 < \left|{x}\right| < \pi$.


Power Series Expansion for Hyperbolic Secant Function

The hyperbolic secant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \sech x       | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}        | c =  }} {{eqn | r = 1 - \frac {x^2} 2 + \frac {5 x^4} {24} - \frac {61 x^6} {720} + \cdots       | c =  }} {{end-eqn}} where $E_{2 n}$ denotes the Euler numbers. This converges for $\size x < \dfrac \pi 2$.


Power Series Expansion for Hyperbolic Cosecant Function

<onlyinclude> The hyperbolic cosecant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \operatorname {csch} x       | r = \sum_{n \mathop = 0}^\infty \dfrac {2 \left({1 - 2^{2 n - 1} }\right) B_{2 n} \,  x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 x - \frac x 6 + \frac {7 x^3} {360} - \frac {31 x^5} {15 \, 120} + \cdots       | c =  }} {{end-eqn}} where $B_n$ denotes the Bernoulli numbers. This converges for $0 < \left\lvert{x}\right\rvert < \pi$.


Sum of Hyperbolic Tangent and Cotangent

:$\tanh x + \coth x = 2 \coth 2 x$


Power Series Expansion for Real Inverse Hyperbolic Sine

The (real) inverse hyperbolic sine function has a Taylor series expansion: :$\sinh^{-1} x = \begin{cases} \displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({-1}\right)^n \left({2 n}\right)!} {2^{2 n} \left({n!}\right)^2} \frac {x^{2 n + 1} } {2 n + 1} & : \left\lvert{x}\right\rvert \le 1 \\  \displaystyle \ln 2 x - \left({\sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^n \left({2 n}\right)!} {2^{2 n + 1} \left({n!}\right)^2 n x^{2 n} } }\right) & : x \ge 1 \\  \displaystyle -\ln \left({-2 x}\right) + \left({\sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^n \left({2 n}\right)!} {2^{2 n + 1} \left({n!}\right)^2 n x^{2 n} } }\right) & : x \le -1 \\  \end{cases}$ That is: :$\sinh^{-1} x = \begin{cases} \displaystyle x - \dfrac 1 2 \dfrac {x^3} 3 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^5} 5 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^7} 7  + \cdots& : \left\lvert{x}\right\rvert \le 1 \\  \displaystyle \ln 2 x - \left({\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\right) & : x \ge 1 \\  \displaystyle -\ln \left({-2 x}\right) + \left({\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\right) & : x \le -1 \\  \end{cases}$


Power Series Expansion for Real Inverse Hyperbolic Cosine

The (real) inverse hyperbolic cosine function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \cosh^{-1} x       | r = \ln 2 x - \left({\sum_{n \mathop = 1}^\infty \frac {\left({2 n}\right)!} {2^{2 n + 1} \left({n!}\right)^2 n x^{2 n} } }\right)       | c =  }} {{eqn | r = \ln 2 x - \left({\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\right)       | c =  }} {{end-eqn}} for $x \ge 1$.


Power Series Expansion for Real Inverse Hyperbolic Tangent

The (real) inverse hyperbolic tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \tanh^{-1} x       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots       | c =  }} {{end-eqn}} for $\size x < 1$.


Power Series Expansion for Real Inverse Hyperbolic Cotangent

The (real) inverse hyperbolic cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \coth^{-1} x       | r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1} x^{2 n + 1} }       | c =  }} {{eqn | r = \frac 1 x + \frac 1 {3 x^3} + \frac 1 {5 x^5} + \frac 1 {7 x^7} + \cdots       | c =  }} {{end-eqn}} for $\size x > 1$.


Power Series Expansion for Exponential of Cosine of x

:$e^{\cos x} = e \left({1 - \dfrac {x^2} 2 + \dfrac {x^4} 6 - \dfrac {31 x^6} {720} + \cdots}\right)$ for all $x \in \R$.


Power Series Expansion for Exponential of Tangent of x

:$e^{\tan x} = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 + \dfrac {3 x^4} 8 + \cdots$ for all $x \in \R$ such that $\size x < \frac \pi 2$.


Power Series Expansion for Exponential of Cosine of x/Proof 2

For all $x \in \R$: :$\displaystyle e^{\cos \left({x}\right)} = e \left({e^{cos \left({x}\right)- 1 } }\right) =  e \left({\sum_{n \mathop = 0}^\infty \frac{ \left({-1}\right)^m P_2 \left({2 m}\right)} {2 m!} x^{2 m} }\right)$ where $P_2 \left({2 m}\right)$ is the partition of the set of size 2m into even blocks.


Sine of x plus Cosine of x

=== Sine Form === {{:Sine of x plus Cosine of x/Sine Form}} === Cosine Form === {{:Sine of x plus Cosine of x/Cosine Form}}


Sine of x plus Cosine of x/Sine Form

:$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$


Cosine of x minus Sine of x

=== Sine Form === {{:Cosine of x minus Sine of x/Sine Form}} === Cosine Form === {{:Cosine of x minus Sine of x/Cosine Form}}


Cosine of x minus Sine of x/Sine Form

:$\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$


Cosine of x minus Sine of x/Cosine Form

:$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$


Power Series Expansion for Exponential of x by Cosine of x

{{begin-eqn}} {{eqn | l = e^x \cos x       | r = \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \cos \left({n \pi / 4}\right) x^n} {n!}       | c =  }} {{eqn | r = 1 + x - \frac {x^3} 3 - \frac {x^4} 6 + \cdots       | c =  }} {{end-eqn}} for all $x \in \R$.


Power Series Expansion for Logarithm of Cosine of x

{{begin-eqn}} {{eqn | l = \ln \size {\cos x}       | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} \{2^{2 n} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}       | c =  }} {{eqn | r = -\frac {x^2} 2 - \frac {x^4} {12} - \frac {x^6} {45} - \frac {17 x^8} {2520} - \cdots       | c =  }} {{end-eqn}} for all $x \in \R$ such that $\size x < \dfrac \pi 2$.


Expectation of Gaussian Distribution

Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then:  :$\expect X = \mu$


Power Series Expansion for Logarithm of Tangent of x

{{begin-eqn}} {{eqn | l = \ln \left\lvert{\tan x}\right\rvert       | r = \ln \left\lvert{x}\right\rvert + \sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^{n - 1} 2^{2 n} \left({2^{2 n - 1} - 1}\right) B_{2 n} \, x^{2 n} } {n \left({2 n}\right)!}       | c =  }} {{eqn | r = \ln \left\lvert{x}\right\rvert + \frac {x^2} 3 + \frac {7 x^4} {90} + \frac {62 x^6} {2835} + \cdots       | c =  }} {{end-eqn}} for all $x \in \R$ such that $0 < \left|{x}\right| < \dfrac \pi 2$.


Derivative of Logarithm over Power

:$\dfrac \d {\d x} \dfrac {\ln x} {x^n} = \dfrac {1 - n \ln x} {x^{n + 1} }$


Nth Derivative of Natural Logarithm by Reciprocal

:$\dfrac {\d^n} {\d x^n} \dfrac {\ln x} x = \paren {-1}^{n + 1} n! \dfrac {H_n - \ln x} {x^{n + 1} }$ where $H_n$ denotes the $n$th harmonic number: :$H_n = \displaystyle \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 r$


Variance of Gaussian Distribution

Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then:  :$\var X = \sigma^2$


Variance as Expectation of Square minus Square of Expectation/Continuous

Let $X$ be a continuous random variable. Then the variance of $X$ can be expressed as: :$\var X = \expect {X^2} - \paren {\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.


Variance as Expectation of Square minus Square of Expectation/Discrete

Let $X$ be a discrete random variable. Then the variance of $X$ can be expressed as: :$\var X = \expect {X^2} - \paren {\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.


Reversion of Power Series

Let $\displaystyle y = \sum_{n \mathop = 1}^\infty c_n x^n$ be a power series. Then: :$\displaystyle x = \sum_{n \mathop = 1}^\infty C_n y^n$ is also a power series, where: {{begin-eqn}} {{eqn | l = c_1 C_1       | r = 1 }} {{eqn | l = {c_1}^3 C_2       | r = -c_2 }} {{eqn | l = {c_1}^5 C_3       | r = 2 {c_2}^2 - c_1 c_3 }} {{eqn | l = {c_1}^7 C_4       | r = 5 c_1 c_2 c_3 - 5 {c_2}^2 - {c_1}^2 c_4 }} {{eqn | l = {c_1}^9 C_5       | r = 6 {c_1}^2 c_2 c_4 + 3 {c_1}^2 {c_3}^2 - {c_1}^3 c_5 + 14 {c_2}^4 - 21 c_1 {c_2}^2 c_3 }} {{eqn | l = {c_1}^{11} C_6       | r = 7 {c_1}^3 c_2 c_5 + 84 c_1 {c_2}^3 c_3 + 7 {c_1}^3 c_3 c_4 - 28 {c_1}^2 c_2 {c_3}^2 - {c_1}^4 c_6 - 28 {c_1}^2 {c_2}^2 c_4 - 42 {c_2}^5 }} {{end-eqn}}


Expectation of Continuous Uniform Distribution

Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.  Then:  :$\expect X = \dfrac {a + b} 2$


Variance of Continuous Uniform Distribution

Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.  Then:  :$\var X = \dfrac {\paren {b - a}^2} {12}$


Median of Gaussian Distribution

Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then the median of $X$ is equal to $\mu$.


Expectation of Gamma Distribution

Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution. The expectation of $X$ is given by: :$\expect X = \dfrac \alpha \beta$


Variance of Gamma Distribution

Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution. The variance of $X$ is given by: :$\var X = \dfrac \alpha {\beta^2}$


Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}       | r = \binom {2 n + 1} 2 2^2 B_2 + \binom {2 n + 1} 4 2^4 B_4 + \binom {2 n + 1} 6 2^6 B_6 + \cdots       | c =  }} {{eqn | r = 2 n       | c =  }} {{end-eqn}} where $B_n$ denotes the $n$th Bernoulli number.


Moment in terms of Moment Generating Function

Let $X$ be a random variable.  Let $M_X$ be the moment generating function of $X$. Then:  :$\expect {X^n} = \map {M^{\paren n}_X} 0$ where: :$n$ is a non-negative integer :$M^{\paren n}_X$ denotes the $n$th derivative of $M_X$ :$\expect {X^n}$ denotes the expectation of $X^n$.


Sum of Euler Numbers by Binomial Coefficient

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = \sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 n - 2 k}       | c =  }} {{eqn | r = \binom {2 n} 2 E_{2 n - 2} + \binom {2 n} 4 E_{2 n - 4} + \binom {2 n} 6 E_{2 n - 6} + \cdots + 1       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number.


Moment Generating Function of Gaussian Distribution

Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$


Bernoulli Number in terms of Euler Numbers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = B_{2 n}       | r = \frac {2n} {2^{2 n} \left({2^{2 n} - 1}\right)} \left({\sum_{k \mathop = 1}^n \dbinom {2 n - 1} {2 k - 1} E_{2 n - 2 k} }\right)       | c =  }} {{eqn | r = \frac {2n} {2^{2 n} \left({2^{2 n} - 1}\right)} \left({\binom {2 n - 1} 1 E_{2 n - 2} + \binom {2 n - 1} 3 E_{2 n - 4} + \binom {2 n - 1} 5 E_{2 n - 6} + \cdots + 1}\right)       | c =  }} {{end-eqn}} where: : $B_n$ denotes the $n$th Bernoulli number : $E_n$ denotes the $n$th Euler number.


Equality of Vector Quantities

Two vector quantities are equal {{iff}} they have the same magnitude and direction. That is: :$\mathbf a = \mathbf b \iff \paren {\size {\mathbf a} = \size {\mathbf b} \land \hat {\mathbf a} = \hat {\mathbf b} }$ where: :$\hat {\mathbf a}$ denotes the unit vector in the direction of $\mathbf a$ :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$.


Vector Addition is Commutative

Let $\mathbf a, \mathbf b$ be vector quantities. Then: :$\mathbf a + \mathbf b = \mathbf b + \mathbf a$


Vector Addition is Associative

Let $\mathbf a, \mathbf b, \mathbf c$ be vectors. Then: :$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$


Scalar Multiplication of Vectors is Associative

Let $\mathbf a$ be a vector quantity. Let $m, n$ be scalar quantities. Then: :$m \paren {n \mathbf a} = \paren {m n} \mathbf a = n \paren {m \mathbf a}$


Scalar Multiplication of Vectors is Distributive over Scalar Addition

Let $\mathbf a$ be a vector quantity. Let $m, n$ be scalar quantities. Then: :$\paren {m + n} \mathbf a = m \mathbf a + n \mathbf a$


Scalar Multiplication of Vectors is Distributive over Vector Addition

Let $\mathbf a, \mathbf b$ be a vector quantities. Let $m$ be a scalar quantity. Then: :$m \paren {\mathbf a + \mathbf b} = m \mathbf a + m \mathbf b$


Scalar Triple Product equals Determinant

Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k }} {{eqn | l = \mathbf b       | r = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k }} {{eqn | l = \mathbf c       | r = c_i \mathbf i + c_j \mathbf j + c_k \mathbf k }} {{end-eqn}} where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of $\mathbf V$. Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$. Then: :$\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right) = \begin{vmatrix} a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \end{vmatrix}$


Differential Entropy of Gaussian Distribution

Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then the differential entropy $\map h X$ of $X$ is given by:  :$\map h X = \map \ln {\sigma \sqrt {2 \pi} } + \dfrac 1 2$


Lagrange's Formula/Corollary

:$\left({\mathbf a \times \mathbf b}\right) \times \mathbf c = \left({\mathbf{a \cdot c} }\right) \mathbf b - \left({\mathbf{b \cdot c} }\right) \mathbf a$


Dot Product of Vector Cross Products

Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3 }} {{eqn | l = \mathbf b       | r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + b_3 \mathbf e_3 }} {{eqn | l = \mathbf c       | r = c_1 \mathbf e_1 + c_2 \mathbf e_2 + c_3 \mathbf e_3 }} {{eqn | l = \mathbf d       | r = d_1 \mathbf e_1 + d_2 \mathbf e_2 + d_3 \mathbf e_3 }} {{end-eqn}} where $\left({\mathbf e_1, \mathbf e_2, \mathbf e_3}\right)$ is the standard ordered basis of $\mathbf V$. Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$. Then: :$\left({\mathbf a \times \mathbf b}\right) \cdot \left({\mathbf c \times \mathbf d}\right) = \left({\mathbf a \cdot \mathbf c}\right) \left({\mathbf b \cdot \mathbf d}\right) - \left({\mathbf a \cdot \mathbf d}\right) \left({\mathbf b \cdot \mathbf c}\right)$


Vector Cross Product of Vector Cross Products

Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3 }} {{eqn | l = \mathbf b       | r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + b_3 \mathbf e_3 }} {{eqn | l = \mathbf c       | r = c_1 \mathbf e_1 + c_2 \mathbf e_2 + c_3 \mathbf e_3 }} {{eqn | l = \mathbf d       | r = d_1 \mathbf e_1 + d_2 \mathbf e_2 + d_3 \mathbf e_3 }} {{end-eqn}} where $\left({\mathbf e_1, \mathbf e_2, \mathbf e_3}\right)$ is the standard ordered basis of $\mathbf V$. Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$. Then: {{begin-eqn}} {{eqn | l = \left({\mathbf a \times \mathbf b}\right) \times \left({\mathbf c \times \mathbf d}\right)       | r = \mathbf c \left({\mathbf a \cdot \left({\mathbf b \times \mathbf d}\right)}\right) - \mathbf d \left({\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right)}\right)       | c =  }} {{eqn | r = \mathbf b \left({\mathbf a \cdot \left({\mathbf c \times \mathbf d}\right)}\right) - \mathbf a \left({\mathbf b \cdot \left({\mathbf c \times \mathbf d}\right)}\right)       | c =  }} {{end-eqn}}


Derivative of Scalar Triple Product of Vector-Valued Functions

Let: : $\mathbf f: x \mapsto \left\langle{f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)}\right\rangle$ : $\mathbf g: x \mapsto \left\langle{g_1 \left({x}\right), g_2 \left({x}\right), \ldots, g_n \left({x}\right)}\right\rangle$ : $\mathbf h: x \mapsto \left\langle{h_1 \left({x}\right), h_2 \left({x}\right), \ldots, h_n \left({x}\right)}\right\rangle$ be differentiable vector-valued functions. The derivative of their scalar triple product is given by: :$\dfrac \d {\d x} \left({\mathbf f \cdot \left({\mathbf g \times \mathbf h}\right)}\right) = \dfrac {\d \mathbf f} {\d x} \cdot \left({\mathbf g \times \mathbf h}\right) + \mathbf f \cdot \left({\dfrac {\d \mathbf g} {\d x} \times \mathbf h}\right) + \mathbf f \cdot \left({\mathbf g \times \dfrac {\d \mathbf h} {\d x} }\right)$


Dot Product of Vector-Valued Function with its Derivative

Let: : $\mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k$ be a differentiable vector-valued function. The dot product of $\mathbf f$ with its derivative is given by: :$\mathbf f \left({x}\right) \cdot \dfrac {\d \mathbf f \left({x}\right)} {\d x} = \left\lvert{\mathbf f \left({x}\right)}\right\rvert \dfrac {\d \left\lvert{\mathbf f \left({x}\right)}\right\rvert} {\d x}$ where $\left\lvert{\mathbf f \left({x}\right)}\right\rvert \ne 0$.


Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero

Let: :$\map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$ be a differentiable vector-valued function. Let $\map {\mathbf f} x$ be such that its magnitude is constant: :$\size {\map {\mathbf f} x} = c$ for some $c \in \R$. Then the dot product of $\mathbf f$ with its derivative is zero: :$\map {\mathbf f} x \cdot \dfrac {\d \map {\mathbf f} x} {\d x} = 0$


Expectation of Beta Distribution

Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ denotes the beta distribution.  Then: :$\expect X = \dfrac \alpha {\alpha + \beta}$


Variance of Beta Distribution

Let $X \sim \map \Beta {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Beta$ is the Beta distribution.  Then: :$\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$


Moment Generating Function of Gamma Distribution

Let $X \sim \Gamma \left({\alpha, \beta}\right)$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution. Then the moment generating function of $X$, $M_X$, is given by:  :$\displaystyle M_X \left({t}\right) = \begin{cases} \left({1 - \frac t \beta}\right)^{-\alpha} & t < \beta \\ \text{does not exist} & t \ge \beta \end{cases}$


Gradient Operator Distributes over Addition

Let $\mathbf V$ be a vector space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$. Let $f \left({x_1, x_2, \ldots, x_n}\right), g \left({x_1, x_2, \ldots, x_n}\right): \mathbf V \to \R$ be real-valued functions on $\mathbf V$. Let $\nabla f$ denote the gradient of $f$. Then: :$\nabla \left({f + g}\right) = \nabla f + \nabla g$


Divergence Operator Distributes over Addition

Let $\mathbf V \left({x_1, x_2, \ldots, x_n}\right)$ be a vector space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$. Let $\mathbf f$ and $\mathbf g: \mathbf V \to \mathbf V$ be vector-valued functions on $\mathbf V$: :$\mathbf f := \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right)$ :$\mathbf g := \left({g_1 \left({\mathbf x}\right), g_2 \left({\mathbf x}\right), \ldots, g_n \left({\mathbf x}\right)}\right)$ Let $\nabla \cdot \mathbf f$ denote the divergence of $f$. Then: :$\nabla \cdot \left({\mathbf f + \mathbf g}\right) = \nabla \cdot \mathbf f + \nabla \cdot \mathbf g$


Curl Operator Distributes over Addition

Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Let $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$: :$\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }$ :$\mathbf g := \tuple {\map {g_x} {\mathbf x}, \map {g_y} {\mathbf x}, \map {g_z} {\mathbf x} }$ Let $\nabla \times \mathbf f$ denote the curl of $f$. Then: :$\nabla \times \paren {\mathbf f + \mathbf g} = \nabla \times \mathbf f + \nabla \times \mathbf g$


Product Rule for Divergence

Let $\mathbf V \left({x_1, x_2, \ldots, x_n}\right)$ be a vector space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$. Let $\mathbf f := \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$. Let $g \left({x_1, x_2, \ldots, x_n}\right): \mathbf V \to \R$ be a real-valued function on $\mathbf V$. Let $\nabla \cdot \mathbf f$ denote the divergence of $f$. Then: :$\nabla \cdot \left({g \, \mathbf f}\right) = g \left({\nabla \cdot \mathbf f}\right) + \left({\nabla g}\right) \cdot \mathbf f$


Product Rule for Curl

Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$. Let $\map g {x, y, z}: \R^3 \to \R$ be a real-valued function on $\R^3$. Let $\nabla \times \mathbf f$ denote the curl of $f$. Then: :$\nabla \times \paren {g \, \mathbf f} = \map g {\nabla \times \mathbf f} + \paren {\nabla g} \times \mathbf f$


Moment Generating Function of Beta Distribution

Let $X \sim \operatorname{Beta} \left({\alpha, \beta}\right)$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the Beta distribution.  Then the moment generating function $M_X$ of $X$ is given by:  :$\displaystyle M_X \left({t}\right) = 1 + \sum_{k \mathop = 1}^\infty \left({ \prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} }\right) \frac{t^k} {k!}$


Raw Moment of Beta Distribution

Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the Beta distribution.  Then: :$\displaystyle \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$ for positive integer $n$.


Linearity of Expectation Function/Discrete

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $\expect X$ denote the expectation of $X$. Then: :$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \, \expect X + \beta \, \expect Y$


Linearity of Expectation Function/Continuous

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $E$ denote the expectation function. Then: :$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \expect X + \beta \expect Y$


Derivative of Constant Multiple/Real

Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then: :$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$


Bound on Complex Values of Gamma Function

Let $\map \Gamma z$ denote the Gamma function. Then for any complex number $z = s + i t$, we have for $\size b \le \size t$: :$\size {\map \Gamma {s + i t} } \le \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$


Set of Vectors defined by Directed Line Segments in Space forms Vector Space

Let $\R^3$ be a real cartesian space of $3$ dimensions. Consider the set $S$ of directed line segments in $\R^3$. Let the equivalence relation $\sim$ be applied to $\R^3$ such that: :$\forall L_1, L_2 \in \R^3: L_1 \sim L_2$ {{iff}} there exists a translation $T$ such that $T \left({L_1}\right) = L_2$ Let $\mathbb V$ denote the set of equivalence classes of $\sim$ on $S$. The elements of $\mathbb V$ form a vector space where: :$\forall \mathbf v_1, \mathbf v_2 \in \mathbb V: \mathbf v_1 + \mathbf v_2$ denotes the element of $\mathbb V$ exemplified by the result of the operation of joining of a representative element of element of $\mathbf v_2$ to the end of a similarly representative element of element of $\mathbf v_1$. :$\forall \mathbf v \in \mathbb V, \lambda \in \R: \lambda \mathbf v$ denotes the element of $\mathbb V$ exemplified by the result of the operation of mulitplying a representative element of element of $\mathbf v$ by the scale factor $\lambda$.


Matrix is Invertible iff Determinant has Multiplicative Inverse/Sufficient Condition

Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Let  the determinant of $\mathbf A$ be invertible in $R$. Then $\mathbf A$ is an invertible matrix.


Chebyshev's Inequality

Let $X$ be a random variable.  Let $\expect X = \mu$ for some $\mu \in \R$.  Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$. Then, for all $k > 0$:  :$\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$


Sign of Permutation on n Letters is Well-Defined

Let $n \in \N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$ be a permutation in $S_n$. Let $\sgn \paren \rho$ denote the sign of $\rho$. Then $\sgn \paren \rho$ is well-defined, in that it is either $1$ or $-1$.


Odd and Even Permutations of Set are Equivalent

Let $n \in \N_{> 0}$ be a natural number greater than $0$. Let $S$ be a set of cardinality $n$. Let $S_n$ denote the symmetric group on $S$ of order $n$. Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively. Then $R_e$ and $R_o$ are equivalent.


Cardinality of Even and Odd Permutations on Finite Set

Let $n \in \N_{> 0}$ be a natural number greater than $0$. Let $S$ be a set of cardinality $n$. Let $S_n$ denote the symmetric group on $S$ of order $n$. Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively. Then the cardinality of both $R_e$ and $R_o$ is $\dfrac {n!} 2$.


Congruence Modulo Integer is Equivalence Relation

For all $z \in \Z$, congruence modulo $z$ is an equivalence relation.


Integer Combination of Coprime Integers/General Result

Let $a_1, a_2, \ldots, a_n$ be integers. Then $\gcd \set {a_1, a_2, \ldots, a_n} = 1$ {{iff}} there exists an integer combination of them equal to $1$: :$\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$


Properties of Beta Function

Let $\Beta \left({x, y}\right)$ denote the Beta function: {{:Definition:Beta Function/Definition 1}} $\Beta \left({x, y}\right)$ has the following properties:


Beta Function of x with y+m+1

Let $\map \Beta {x, y}$ denote the Beta function. Then: :$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$ where $\Gamma_m$ is the partial Gamma function: :$\map {\Gamma_m} y := \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \cdots \paren {y + m} }$


Binomial Coefficient expressed using Beta Function

Let $\dbinom r k$ denote a binomial coefficient. Then: :$\dbinom r k = \dfrac 1 {\left({r + 1}\right) B \left({k + 1, r - k + 1}\right)}$


Symmetry Rule for Binomial Coefficients/Complex Numbers

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\dbinom z w = \dbinom z {z - w}$


Factors of Binomial Coefficient/Complex Numbers

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\dbinom z w = \dfrac z w \dbinom {z - 1} {w - 1}$ where $\dbinom z w$ is a binomial coefficient.


Pascal's Rule/Complex Numbers

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\dbinom z {w - 1} + \dbinom z w = \dbinom {z + 1} w$ where $\dbinom z w$ is a binomial coefficient.


Binomial Theorem/Extended

Let $r, \alpha \in \C$ be complex numbers. Let $z \in \C$ be a complex number such that $\left|{z}\right| < 1$. Then: :$\displaystyle \paren {1 + z}^r = \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} z^{\alpha + k}$ where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.


Chu-Vandermonde Identity/Extended

Let $r, s, \alpha, \beta \in \C$ be complex numbers. Then: :$\displaystyle \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} \dbinom s {\beta - k}$ where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.


Beta Function of Half with Half

:$\Beta \left({\dfrac 1 2, \dfrac 1 2}\right) = \pi$ where $\Beta$ denotes the Beta function.


Binomial Coefficient of Real Number with Half

:$\dbinom r {1 / 2} = \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}$ where $\dbinom r {1 / 2}$ denotes a binomial coefficient.


Limit to Infinity of Binomial Coefficient over Power

:$\displaystyle \lim_{r \mathop \to \infty} \dfrac {\dbinom r k} {r^k} = \frac 1 {\map \Gamma {k + 1}}$


Approximation to x+y Choose y

:$\displaystyle \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$


Product of r Choose k with r Minus Half Choose k/Formulation 1

Let $k \in \Z$, $r \in \R$. :$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$ where $\dbinom r k$ denotes a binomial coefficient.


Provable Consequence of Theorems is Theorem

Let $\mathcal P$ be a proof system for a formal language $\mathcal L$. Let $\mathcal F$ be a collection of theorems of $\mathcal P$. Denote with $\mathscr P \left({\mathcal F}\right)$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\mathcal F$ as axioms. Let $\phi$ be a provable consequence of $\mathcal F$: :$\vdash_{\mathscr P} \mathcal F$ :$\mathcal F \vdash_{\mathscr P} \phi$ Then $\phi$ is also a theorem of $\mathscr P$: :$\vdash_{\mathscr P} \phi$


Binomial Coefficient of Minus Half

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


Falling Factorial of Sum of Integers

Let $r \in \R$ be a real number. Let $a, b \in \Z$ be (positive) integers. Then: :$r^{\underline {a + b} } = r^{\underline a} \left({r - a}\right)^{\underline b}$ where $r^{\underline a}$ denotes the $a$th falling factorial of $r$.


Approximate Size of Sum of Harmonic Series

Let $H_n$ denote the sum of the harmonic series: :$H_n = \displaystyle \sum_{k \mathop = 1}^n \frac 1 k$ Then $H_n$ can be approximated as follows: :$H_n \approx \ln n + \gamma + \dfrac 1 {2 n} - \dfrac 1 {12 n^2} + \dfrac 1 {120 n^4} - \epsilon$ where: :$\gamma$ denotes the Euler-Mascheroni constant: $\gamma \approx 0 \cdotp 57721 \, 56649 \, \ldots$ :$0 < \epsilon < \dfrac 1 {252 n^6}$


Riemann Zeta Function of 8

The Riemann zeta function of $8$ is given by: {{begin-eqn}} {{eqn | l = \map \zeta 8       | r = \dfrac 1 {1^8} +  \dfrac 1 {2^8} +  \dfrac 1 {3^8} +  \dfrac 1 {4^8} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^8} {9450}       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 00408 \, 3 \ldots       | c =  }} {{end-eqn}}


Sum of Sequence of Harmonic Numbers

:$\displaystyle \sum_{k \mathop = 1}^n H_k = \left({n + 1}\right) H_n - n$ where $H_k$ denotes the $k$th harmonic number.


Sum over k to n of k Choose m by kth Harmonic Number

:$\displaystyle \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \left({H_{n + 1} - \frac 1 {m + 1} }\right)$ where: : $\dbinom k m$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number.


Sum over k of n Choose k by x to the k by kth Harmonic Number

Let $x \in \R_{> 0}$ be a real number. Then: :$\displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k = \left({x + 1}\right)^n \left({H_n - \ln \left({1 + \frac 1 x}\right)}\right) + \epsilon$ where: : $\dbinom n k$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number : $0 < \epsilon < \dfrac 1 {x \left({n + 1}\right)}$


Power to Real Number by Decimal Expansion is Uniquely Defined

Let $r \in \R_{> 1}$ be a real number greater than $1$, expressed by its decimal expansion: :$r = n \cdotp d_1 d_2 d_3 \ldots$ The power $x^r$ of a (strictly) positive real number $x$ defined as: :$(1): \quad \displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$ where: {{begin-eqn}} {{eqn | l = \psi_1       | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {10^k} = n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k} }} {{eqn | l = \psi_2       | r = \psi_1 + \dfrac 1 {10^k} }} {{end-eqn}} is unique.


Change of Base of Logarithm/Base 2 to Base 10

:$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right) = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots \lg x$


Ordering on Real Numbers from Decimal Expansion

Let $x, y \in \R$ be real numbers. Let $x$ and $y$ be expressed by their decimal expansions: {{begin-eqn}} {{eqn | l = x       | r = m \cdotp d_1 d_2 d_3 \ldots }} {{eqn | l = y       | r = n \cdotp e_1 e_2 e_3 \ldots       | c =  }} {{end-eqn}} Let $\preccurlyeq$ be the lexicographic ordering on $\R$ defined as: :$x \preccurlyeq y$ {{iff}}: ::$m \prec n$ :or: ::$m = n$ and $\exists k \in \Z_{>0}: \left({\forall j: 1 \le j < k: d_j = e_j}\right) \land d_k < e_k$ :or: ::$m = n$ and $\forall j \in \Z_{>0}: d_j = e_j$. Then: : $x \le y$ where $\le$ denotes the usual ordering on $\R$.


Necessary Precision for x equal to log base 10 of 2 to determine Decimal expansion of 10 to the x

Let $b = 10$. Let $x \approx \log_{10} 2$. Let it be necessary to calculate the decimal expansion of $x$ to determine the first $3$ decimal places of $b^x$. An infinite number of decimal places of $x$ would in fact be necessary.


Change of Base of Logarithm/Base 10 to Base e/Form 1

:$\ln x = \paren {\ln 10} \paren {\log_{10} x} = 2 \cdotp 30258 \, 50929 \, 94 \ldots \log_{10} x$


Change of Base of Logarithm/Base 10 to Base e/Form 2

:$\ln x = \dfrac {\log_{10} x} {\log_{10} e} = \dfrac {\log_{10} x} {0 \cdotp 43429 \, 44819 \, 03 \ldots}$


Change of Base of Logarithm/Base e to Base 10/Form 1

:$\log_{10} x = \paren {\log_{10} e} \paren {\ln x} = 0 \cdotp 43429 \, 44819 \, 03 \ldots \ln x$


Change of Base of Logarithm/Base e to Base 10/Form 2

:$\log_{10} x = \dfrac {\ln x} {\ln 10} = \dfrac {\ln x} {2 \cdotp 30258 \, 50929 \, 94 \ldots}$


Change of Base of Logarithm/Base 10 to Base 2

:$\lg x = \dfrac {\log_{10} x} {\log_{10} 2} = \dfrac {\log_{10} x} {0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots}$


Root of Quotient equals Quotient of Roots

:$\sqrt [n] {\dfrac a b} = \dfrac {\sqrt [n] a} {\sqrt [n] b}$


Logarithm to Own Base equals 1

Let $b \in \R_{>0}$ be a strictly positive real number such that $b \ne 1$. Let $\log_b$ denote the logarithm to base $b$. Then: :$\log_b b = 1$


General Logarithm/Examples/Base b of 1

:$\log_b 1 = 0$


General Logarithm/Examples/Base b of -1

:$\log_b \left({-1}\right)$ is undefined in the real number line.


Change of Base of Logarithm/Base 2 to Base 8

:$\log_8 x = \dfrac {\lg x} 3$


Number of Digits to Represent Integer in Given Number Base

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $b \in \Z$ be an integer such that $b > 1$. Let $d$ denote the number of digits of $n$ when represented in base $b$. Then: :$d = \ceiling {\map {\log_b} {n + 1} }$ where $\ceiling {\, \cdot \,}$ denotes the ceiling function.


Number of Bits for Decimal Integer

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $n$ have $m$ digits when expressed in decimal notation. Then $n$ may require as many as $\ceiling {\dfrac m {\log_{10} 2} }$ bits to represent it.


Sum of Summations over Overlapping Domains/Infinite Series

Let the fiber of truth of both $R$ and $S$ be infinite. Then provided that any $3$ of the $4$ summations converge: :$\ds \sum_{\map R j} a_j + \sum_{\map S j} a_j = \sum_{\map R j \mathop \lor \map S J} a_j + \sum_{\map R j \mathop \land \map S j} a_j$ where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.


Translation of Index Variable of Summation/Infinite Series

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the set of integers. Let $\displaystyle \sum_{R \left({j}\right)} a_j$ denote a summation over $R$. Let the fiber of truth of $R$ be infinite. Then: :$\displaystyle \sum_{R \left({j}\right)} a_j = \sum_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \sum_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$ where $c$ is an integer constant which is not dependent upon $j$.


Exchange of Order of Summation/Example

Let the fiber of truth of both $R$ and $S$ be infinite. Then it is not necessarily the case that: :$\displaystyle \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$


Sum of Geometric Sequence/Examples/Index to Minus 1

Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$. Then the formula for Sum of Geometric Sequence: :$\displaystyle \sum_{j \mathop = 0}^n x^j = \frac {x^{n + 1} - 1} {x - 1}$ still holds when $n = -1$: :$\displaystyle \sum_{j \mathop = 0}^{-1} x^j = \frac {x^0 - 1} {x - 1}$


Sum of Geometric Sequence/Examples/Common Ratio 1

Consider the Sum of Geometric Sequence defined on the standard number fields for all $x \ne 1$. :$\displaystyle \sum_{j \mathop = 0}^n a x^j = a \paren {\frac {1 - x^{n + 1} } {1 - x} }$ When $x = 1$, the formula reduces to: :$\displaystyle \sum_{j \mathop = 0}^n a 1^j = a \paren {n + 1}$


Sum of Geometric Sequence/Examples/One Seventh from 1 to n

:$\displaystyle \sum_{j \mathop = 0}^n \dfrac 1 {7^j} = \frac 7 6 \left({1 - \frac 1 {7^{n + 1} } }\right)$


Summation of Unity over Elements

Let $S \subseteq \Z$ be a set of integers. Let: :$n := \displaystyle \sum_{j \mathop \in  S} 1$ Then $n$ is equal to the cardinality of $S$.


Hausdorff Maximal Principle Implies Zorn's Lemma

The Hausdorff Maximal Principle implies Zorn's Lemma.


Zorn's Lemma Implies Well Ordering Theorem

Zorn's Lemma implies the Well-Ordering Theorem.


Uncountable Sum as Series

Let $X$ be an uncountable set. Let $f: X \to \closedint 0 {+\infty}$ be an extended real-valued function. The uncountable sum: :$\displaystyle \sum_{x \mathop \in X} \map f x = \sup \set {\sum_{x \mathop \in F} \map f x : F \subseteq X, F \text{ finite} }$ is $+\infty$, or can be expressed as a (possibly divergent) series.


Sum with Maximum is Maximum of Sum

Let $a, b, c \in \R$ be real numbers. Then: :$a + \max \set {b, c} = \max \set {a + b, a + c}$


Sum of Indexed Suprema

Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$. Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the real numbers $\R$ indexed by $J$. Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively. Then: :$\displaystyle \left({\sup_{R \left({i}\right)} a_i}\right) + \left({\sup_{S \left({j}\right)} b_j}\right) = \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} \left({a_i + b_j}\right)}\right)$


Product of Indexed Suprema of Non-Negative Numbers

Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$. Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively. Then: :$\displaystyle \left({\sup_{R \left({i}\right)} a_i}\right) \left({\sup_{S \left({j}\right)} b_j}\right) = \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} \left({a_i b_j}\right)}\right)$


Change of Index Variable of Supremum

Let $\family {a_i}{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\map R i$ be a propositional functions of $i \in I$. Let $\displaystyle \sup_{\map R i} a_i$ be the indexed supremum on $\family {a_i}$. Then: :$\displaystyle \sup_{\map R i} a_i = \sup_{\map R j} a_j$


Permutation of Indices of Supremum

Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $R \left({i}\right)$ be a propositional functions of $i \in I$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ be the indexed supremum on $\left \langle {a_i} \right \rangle$. Then: :$\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({\pi \left({i}\right)}\right)} a_{\pi \left({i}\right)}$ where $\pi$ is a permutation on the fiber of truth of $R$.


Exchange of Order of Supremum Operators

Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$. Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively. Then: :$\displaystyle \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} a_{i j} }\right) = \sup_{S \left({j}\right)} \left({\sup_{R \left({i}\right)} a_{i j} }\right)$


Supremum of Suprema over Overlapping Domains

Let $\family {a_i}_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\map R i$ and $\map S i$ be propositional functions of $i \in I$. Let $\displaystyle \sup_{\map R i} a_i$ and $\displaystyle \sup_{\map S i} a_i$ be the indexed suprema on $\family {a_i}$ over $\map R i$ and $\map S i$ respectively. Then: :$\displaystyle \map \sup {\sup_{\map R i} a_i, \sup_{\map S i} a_i} = \sup_{\map R i \mathop \lor \map S i} a_i$


Contour Integration by Substitution

Let $f$ be a holomorphic function on a simply connected domain $V \subseteq \mathbb C$. Let $\gamma$ be a contour in $V$ starting at $z_1$ and ending at $z_2$. Let $U$ be a connected domain. Let $\phi: U \to V$ be a holomorphic function with $\phi^{-1} \sqbrk {\set {z_1, z_2} } \ne \O$. Let $\omega$ be a contour in $U$ starting at $u_1$ and ending at $u_2$, such that $\map \phi {u_1} = z_1$ and $\map \phi {u_2} = z_2$. Then the contour integral of $f$ over $\gamma$ satisfies the following substitution: :$\displaystyle \int_\gamma \map f z \rd z = \int_\omega \map f {\map \phi u} \, \map {\phi'} u \rd u$


Infimum of Set of Integers is Integer

Let $S \subset \Z$ be a non-empty subset of the set of integers. Let $S$ be bounded below in the set of real numbers. Then its infimum $\inf S$ is an integer.


Infimum of Set of Integers equals Smallest Element

Let $S \subset \Z$ be a non-empty subset of the set of integers. Let $S$ be bounded below in the set of real numbers $\R$. Then $S$ has a smallest element, and it is equal to the infimum $\sup S$.


Greatest Element is Supremum

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. Let $T$ have a greatest element $M$. Then $M$ is the supremum of $T$ in $S$.


Smallest Element is Infimum

Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $T$ have a smallest element $m$. Then $m$ is the infimum of $T$ in $S$.


Floor Function/Examples/Floor of 1.1

:$\floor {1 \cdotp 1} = 1$


Floor Function/Examples/Floor of -1.1

:$\floor {-1 \cdotp 1} = -2$


Ceiling Function/Examples/Ceiling of -1.1

:$\ceiling {-1 \cdotp 1} = -1$


Floor Function/Examples/Floor of 0.99999

:$\floor {0 \cdotp 99999} = 0$


Floor Function/Examples/Floor of Binary Logarithm of 35

:$\floor {\lg 35} = 5$ where: : $\lg x$ denotes the binary logarithm ($\log_2$) of $x$


Modulo Operation/Examples/100 mod 3

:$100 \bmod 3 = 1$


Modulo Operation/Examples/-100 mod 7

:$-100 \bmod 7 = 5$


Modulo Operation/Examples/-100 mod 0

:$-100 \bmod 0 = -100$


Modulo Operation/Examples/5 mod -3

:$5 \bmod -3 = 5$


Modulo Operation/Examples/18 mod -3

:$18 \bmod -3 = 0$


Modulo Operation/Examples/-2 mod -3

:$-2 \bmod -3 = -2$


Modulo Operation/Examples/0.11 mod -0.1

:$0 \cdotp 11 \bmod -0 \cdotp 1 = -0 \cdotp 09$


Common Factor Cancelling in Congruence/Corollary 1/Warning

Let $a$ ''not'' be coprime to $m$. Then it is not necessarily the case that: :$x \equiv y \pmod m$


Chinese Remainder Theorem/Warning

Let $r$ ''not'' be coprime to $s$. Then it is not necessarily the case that: :$a \equiv b \pmod {r s}$ {{iff}} $a \equiv b \pmod r$ and $a \equiv b \pmod s$ where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.


Euler Phi Function of 2

:$\map \phi 2 = 1$


Existence of Abscissa of Convergence

Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$ be a Dirichlet series. Let the series $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n n^{-s} }$ not converge for all $s \in \C$, or diverge for all $s \in \C$. Then there exists a real number $\sigma_c$ such that $\map f s$ converges for all $s = \sigma + it$ with $\sigma > \sigma_c$, and does not converge for all $s$ with $\sigma < \sigma_c$. We call $\sigma_c$ the '''abscissa of convergence''' of the Dirichlet series.