hcju/proofwikips huggingface dataset:
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.  :$(1): \quad A$ 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.  :$(1): \quad \mathfrak a$ 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.  :$(1): \quad$ 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.  # $I$ 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 $\closedint a b$.