hcju/proofwikips huggingface dataset:
Primitive of Function of Nth Root of a x + b
:$\displaystyle \int F \left({\sqrt [n] {a x + b}}\right) \ \mathrm d x = \frac n a \int u^{n-1} F \left({u}\right) \ \mathrm d u$ where $u = \sqrt [n] {a x + b}$.
Primitive of Function of Root of a squared plus x squared
:$\displaystyle \int F \left({\sqrt {a^2 + x^2}}\right) \ \mathrm d x = a \int \sec^2 u \ F \left({a \sec u}\right) \ \mathrm d u$ where $x = a \tan u$.
Primitive of Function of Natural Logarithm
:$\displaystyle \int F \left({\ln x}\right) \rd x = \int F \left({u}\right) e^u \rd u$ where $u = \ln x$.
Primitive of Function of Arccosecant
:$\displaystyle \int F \left({\operatorname{arccsc} \frac x a}\right) \ \mathrm d x = -a \int F \left({u}\right) \left\vert{\csc u}\right\vert \cot u \ \mathrm d u$ where $u = \operatorname{arccsc} \dfrac x a$.
Subtraction of Subring is Subtraction of Ring
Let $\struct {R, +, \circ}$ be an ring. For each $x, y \in R$ let $x - y$ denote the subtraction of $x$ and $y$ in $R$. Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$. For each $x, y \in S$ let $x \sim y$ denote the subtraction of $x$ and $y$ in $S$. Then: :$\forall x, y \in S: x \sim y = x - y$
Area between Smooth Curve and Line is Maximized by Semicircle
Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space. Let $y$ have a total length of $l$. Let it be contained in the upper half-plane with an exception of endpoints, which are on the $x$-axis. Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area. Then $y$ is a semicircle.
Unique Representation in Polynomial Forms
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $X \in R$ be transcendental over $D$. Let $D \sqbrk X$ be the ring of polynomials in $X$ over $D$. Then each non-zero member of $D \left[{X}\right]$ can be expressed in just one way in the form: :$\ds f \in D \sqbrk X: f = \sum_{k \mathop = 0}^n {a_k \circ X^k}$
Rings of Polynomials in Ring Elements are Isomorphic
Let $R_1, R_2$ be commutative rings with unity. Let $D$ be an integral subdomain of both $R_1$ and $R_2$. Let $X_1, X_2 \in R$ be transcendental over $D$. Let $D \sqbrk {X_1}, D \sqbrk {X_2}$ be the rings of polynomials in $X_1$ and $X_2$ over $D$. Then $D \sqbrk {X_1}$ is isomorphic to $D \sqbrk {X_2}$.
Laplace Transform of Higher Order Derivatives
{{begin-eqn}} {{eqn | l = \laptrans {\map {f^{\paren n} } t}       | r = s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0 }} {{eqn | r = s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f''} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \map {f^{\paren {n - 1} } } 0 }} {{end-eqn}}
Laplace Transform of Error Function
:$\laptrans {\map \erf t} = \dfrac 1 s \, \map \exp {\dfrac {s^2} 4} \, \map \erfc {\dfrac s 2}$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\erf$ denotes the error function :$\erfc$ denotes the complementary error function :$\exp$ denotes the exponential function.
Laplace Transform of Exponential Integral Function
:$\laptrans {\map \Ei t} = \dfrac {\map \ln {s + 1} } s$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Ei$ denotes the exponential integral function.
Laplace Transform of Shifted Dirac Delta Function
Let $\map \delta t$ denote the Dirac delta function. The Laplace transform of $\map \delta {t - a}$ is given by: :$\laptrans {\map \delta {t - a} } = e^{-a s}$
Primitive of x over Root of a x + b
:$\displaystyle \int \frac {x \ \mathrm d x} {\sqrt{a x + b} } = \frac {2 \left({a x - 2 b}\right) \sqrt{a x + b} } {3 a^2}$
Laplace Transform of t by Sine a t
Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {t \sin a t} = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$
Laplace Transform of Sine of t over t/Corollary
:$\laptrans {\dfrac {\sin a t} t} = \arctan \dfrac a s$
Laplace Transform of Natural Logarithm
:$\laptrans {\ln t} = \dfrac {\map {\Gamma'} 1 - \ln s} s = -\dfrac {\gamma + \ln s} s$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Gamma$ denotes the Gamma function :$\gamma$ denotes the Euler-Mascheroni constant.
Convolution Theorem
Let $\mathbb F \in \set {\R, \C}$.  Let $f: \R \to \F$ and $g: \R \to \F$ be functions. Let their Laplace transforms $\laptrans {\map f t} = \map F s$ and $\laptrans {\map g t} = \map G s$ exist. Then: :$\map F s \map G s = \displaystyle \laptrans {\int_0^t \map f u \map g {t - u} \rd u}$
Primitive of p x + q over Root of a x + b
:$\displaystyle \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$
Negative of Supremum is Infimum of Negatives
Let $S$ be a subset of the real numbers $\R$. Let $S$ be bounded above. Then: :$(1): \quad \set {x \in \R: -x \in S}$ is bounded below :$(2): \quad \displaystyle -\sup_{x \mathop \in S} x = \map {\inf_{x \mathop \in S} } {-x}$ where $\sup$ and $\inf$ denote the supremum and infimum respectively.
Propagation of Light in Inhomogeneous Medium
Let $v: \R^3 \to \R$ be a real function. Let $M$ be a 3-dimensional Euclidean space. Let $\gamma:t \in \R \to M$ be a smooth curve embedded in $M$, where $t$ is time. Denote its derivative  time by $v$. Suppose $M$ is filled with an optically inhomogeneous medium such that at each point speed of light is $v = \map v {x, y, z}$ Suppose $\map y x$ and $\map z x$ are real functions. Let the light move according to Fermat's principle. Then equations of motion have the following form: :$\dfrac {\partial v} {\partial y} \dfrac {\sqrt {1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \dfrac {y'} {v \sqrt {1 + y'^2 + z'^2} } = 0$ :$\dfrac {\partial v} {\partial z} \dfrac {\sqrt {1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \dfrac {z'} {v \sqrt {1 + y'^2 + z'^2} } = 0$
Primitive of Reciprocal of x by Power of x squared minus a squared
:$\displaystyle \int \frac {\mathrm d x} {x \left({x^2 - a^2}\right)^n} = \frac {-1} {2 \left({n - 1}\right) a^2 \left({x^2 - a^2}\right)^{n - 1} } - \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({x^2 - a^2}\right)^{n - 1} }$ for $x^2 > a^2$.
Primitive of Reciprocal of Power of x by Power of x squared minus a squared
:$\displaystyle \int \frac {\d x} {x^m \paren {x^2 - a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }$ for $x^2 > a^2$.
Signum Function is Quotient of Number with Absolute Value
Let $x \in \R_{\ne 0}$ be a non-zero real number. Then: :$\operatorname{sgn} \left({x}\right) = \dfrac x {\left\vert{x}\right\vert} = \dfrac {\left\vert{x}\right\vert} x$ where: :$\operatorname{sgn} \left({x}\right)$ denotes the signum function of $x$ :$\left\vert{x}\right\vert$ denotes the absolute value of $x$.
Local Basis Generated from Neighborhood Basis
Let $T = \struct {S, \tau}$ be a topological space. Let $x$ be an element of $S$. Let $\BB$ be a neighborhood basis of $x$. For any subset $A \subseteq S$, let $A^\circ$ denote the interior of $A$. Then the set: :$\BB' = \set {H^\circ: H \in B}$ is a local basis of $x$.
Excess Kurtosis of Bernoulli Distribution
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac {1 - 6 p q} {p q}$ where $q = 1 - p$.
Skewness of Beta Distribution
Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the Beta distribution.  Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }$
Primitive of x over Power of a squared minus x squared
:$\displaystyle \int \frac {x \rd x} {\paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} \paren {a^2 - x^2}^{n - 1} }$ for $x^2 < a^2$.
Primitive of Reciprocal of x by Power of a squared minus x squared
:$\displaystyle \int \frac {\mathrm d x} {x \left({a^2 - x^2}\right)^n} = \frac 1 {2 \left({n - 1}\right) a^2 \left({a^2 - x^2}\right)^{n - 1} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({a^2 - x^2}\right)^{n - 1} }$ for $x^2 < a^2$.
Primitive of x squared by Root of x squared plus a squared
:$\displaystyle \int x^2 \sqrt {x^2 + a^2} \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 - \frac {a^2 x \sqrt {x^2 + a^2} } 8 - \frac {a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$
Sum of Squares of Standard Gaussian Random Variables has Chi-Squared Distribution
Let $X_1, X_2, \ldots, X_n$ be independent random variables. Let $X_i \sim \Gaussian 0 1$ for $1 \le i \le n$ where $\Gaussian 0 1$ is the standard Gaussian Distribution. Then:  :$\displaystyle \sum_{i \mathop = 1}^n X^2_i \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Primitive of x squared over Root of x squared plus a squared cubed
:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({\sqrt {x^2 + a^2} }\right)^3} = \frac {-x} {\sqrt {x^2 + a^2} } + \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$
Primitive of Reciprocal of x squared by Root of x squared plus a squared cubed
:$\displaystyle \int \frac {\d x} {x^2 \paren {\sqrt {x^2 + a^2} }^3} = \frac {-\sqrt {x^2 + a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 + a^2} } + C$
Variance of Student's t-Distribution
Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the variance of $X$ is given by:  :$\var X = \dfrac k {k - 2}$ for $k > 2$, and does not exist otherwise.
Primitive of Root of x squared plus a squared cubed over x
:$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + a^2 \sqrt {x^2 + a^2} - a^3 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
Primitive of Root of x squared plus a squared cubed over x squared
:$\displaystyle \int \frac{\left({\sqrt {x^2 + a^2} }\right)^3} {x^2} \ \mathrm d x = \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} x + \frac{3 x \sqrt {x^2 + a^2} } 2 + \frac {3 a^2} 2 \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$
Weak Whitney Immersion Theorem
Every $k$-dimensional manifold $X$ admits a one-to-one immersion in $\R^{2 k + 1}$. {{MissingLinks|$k$-dimensional}}
Primitive of Root of x squared plus a squared cubed over x cubed
:$\displaystyle \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
Homotopy Group is Homeomorphism Invariant
Let $X$ and $Y$ be two topological spaces. Let $\phi: X \to Y$ be a homeomorphism. Let $x_0 \in X$, $y_0 \in Y$. Then for all $n \in \N$ the induced mapping: :$\phi_* : \pi_n \left({X, x_0}\right) \to \pi_n \left({Y, y_0}\right):$ ::$\left[{\!\left[{\, c \,}\right]\!}\right] \mapsto \left[{\!\left[{\, \phi \circ c \,}\right]\!}\right]$ is an isomorphism, where $\pi_n$ denotes the $n$th homotopy group.
Primitive of x cubed by Root of x squared minus a squared
:$\displaystyle \int x^3 \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^3} 3 + C$
Primitive of Reciprocal of x squared by Root of x squared minus a squared cubed
:$\displaystyle \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3} = \frac {-\sqrt {x^2 - a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 - a^2} } + C$
Primitive of x squared by Root of x squared minus a squared cubed
:$\displaystyle \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {x^2 - a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 - a^2} } {16} + \frac {a^6} {16} \ln \size {x + \sqrt {x^2 - a^2} } + C$
Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n
Let $\dbinom {2 n} n$ denote a binomial coefficient. Then for all prime numbers $p$ such that $n < p < 2 n$: :$p \divides \dbinom {2 n} n$ where $\divides$ denotes divisibility.
Linear Combination of Gaussian Random Variables
Let $X_1, X_2, X_3, \ldots, X_n$ be independent random variables.  Let $\sequence {\alpha_i}_{1 \le i \le n}$ and $\sequence {\mu_i}_{1 \le i \le n}$ be sequences of real numbers. Let $\sequence {\sigma_i}_{1 \le i \le n}$ be a sequence of positive real numbers. Let $X_i \sim \Gaussian {\mu_i} {\sigma^2_i}$ for $1 \le i \le n$, where $\Gaussian {\mu_i} {\sigma^2_i}$ is the Gaussian distribution with parameters $\mu_i$ and $\sigma^2_i$. Then:  :$\displaystyle \sum_{i \mathop = 1}^n \alpha_i X_i \sim \Gaussian {\sum_{i \mathop = 1}^n \alpha_i \mu_i} {\sum_{i \mathop = 1}^n \alpha^2_i \sigma^2_i}$
Primitive of x squared by Root of a squared minus x squared
:$\displaystyle \int x^2 \sqrt {a^2 - x^2} \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \sinh^{-1} \frac x a + C$
Existence of Non-Measurable Subset of Real Numbers
There exists a subset of the real numbers which is not measurable.
Cube as Difference between Two Squares
A cube number can be expressed as the difference between two squares.
Factorial Greater than Cube for n Greater than 5
Let $n \in \Z$ be an integer such that $n > 5$. Then $n! > n^3$.
Existence of Euler-Mascheroni Constant
The real sequence: :$\displaystyle \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$ converges to a limit. This limit is known as the Euler-Mascheroni constant.
Square Product of Three Consecutive Triangular Numbers
Let $T_n$ denote the $n$th triangular number for $n \in \Z_{>0}$ a (strictly) positive integer. Let $T_n \times T_{n + 1} \times T_{n + 2}$ be a square number. Then at least one value of $n$ fulfils this condition: :$n = 3$
Cube Modulo 9
Let $x \in \Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^3       | o = \equiv       | r = 0 \pmod 9       | c =  }} {{eqn | l = x^3       | o = \equiv       | r = 1 \pmod 9       | c =  }} {{eqn | l = x^3       | o = \equiv       | r = 8 \pmod 9       | c =  }} {{end-eqn}}
Acceleration of Particle moving in Circle
Let $P$ be a particle moving in a circular path $C$. Then the acceleration of $P$ is given as: :$\mathbf a = -\dfrac {\size {\mathbf v}^2 \mathbf r} {\size {\mathbf r}^2}$ where: :$\mathbf v$ is the instantaneous velocity of $P$ :$\mathbf r$ is the vector whose magnitude equals the length of the radius of $C$ and whose direction is from the center of $C$ to $P$ :$\size {\, \cdot \,}$ denotes the magnitude of a vector.
Primitive of Reciprocal of Root of a x squared plus b x plus c/a equal to 0
:$\displaystyle \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} } = \frac {2 \sqrt {b x + c} } b + C$ when $a = 0$.
Primitive of x squared over Cube of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {\left({2 b^2 - 4 a c}\right) x + 2 b c} {a \left({4 a c - b^2}\right) \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} }$
Primitive of Reciprocal of x squared by Cube of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} = -\frac {a x^2 + 2 b x + c} {c^2 x \sqrt {a x^2 + b x + c} } + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac {3 b} {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$
Dirichlet Beta Function at Odd Positive Integers
:$\map \beta {2 n + 1} = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$
Primitive of Reciprocal of Half Integer Power of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^{n + \frac 1 2} } = \frac {2 \left({2 a x + b}\right)} {\left({2 n - 1}\right) \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)^{n - \frac 1 2} } + \frac {8 a \left({n - 1}\right)} {\left({2 n - 1}\right) \left({4 a c - b^2}\right)} \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^{n - \frac 1 2} }$
Primitive of x squared over x cubed plus a cubed
:$\displaystyle \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$
Primitive of x over x cubed plus a cubed squared
:$\displaystyle \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)^2} = \frac {x^2} {3 a^3 \left({x^3 + a^3}\right)} + \frac 1 {18 a^4} \ln \left({\frac {x^2 - a x + a^2} {\left({x + a}\right)^2} }\right) + \frac 1 {3 a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$
Compact First-Countable Space is Sequentially Compact
Let $T = \struct {S, \tau}$ be a compact first-countable topological space. Then every infinite sequence in $S$ has a convergent subsequence; that is, $T$ is sequentially compact.
Convergence in Indiscrete Space
Let $\left({S, \left\{{S, \varnothing}\right\}}\right)$ be an indiscrete space. Let $\left \langle {x_n} \right \rangle$ be any sequence in $S$. Then $\left \langle {x_n} \right \rangle$ converges to any point $x$ of $S$.
Möbius Transformation is Bijection/Restriction to Reals
Let $a, b, c, d \in \R$ be real numbers. Let $f: \R^* \to \R^*$ be the Möbius transformation restricted to the real numbers: :$\map f x = \begin {cases} \dfrac {a x + b} {c x + d} & : x \ne -\dfrac d c \\ \infty & : x = -\dfrac d c \\ \dfrac a c & : x = \infty \\ \infty & : x = \infty \text { and } c = 0 \end {cases}$ Then: :$f: \R^* \to \R^*$ is a bijection : :$a c - b d \ne 0$
Definite Integral from 0 to 1 of Logarithm of x over One plus x
:$\displaystyle \int_0^1 \frac {\ln x} {1 + x} \rd x = -\frac {\pi^2} {12}$
Primitive of Reciprocal of x by Root of Power of x minus Power of a
:$\displaystyle \int \frac {\d x} {x \sqrt {x^n - a^n} } = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }$
Set of Sets can be Defined as Family
Let $\Bbb S$ be a set of sets. Then $\Bbb S$ can be defined as an indexed family of sets.
Primitive of Sine of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\sin a x \d x} x       | r = \sum_{k \mathop \ge 0} \frac {\paren {-1}^k \paren {a x}^{2 k + 1} } {\paren {2 k + 1} \paren {2 k + 1}!} + C }} {{eqn | r = a x - \frac {\paren {a x}^3} {3 \times 3!} + \frac {\paren {a x}^5} {5 \times 5!} - \cdots }} {{end-eqn}}
Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form
:$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\csc a x - \cot a x} + C$
Primitive of Reciprocal of Square of 1 minus Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\left({1 - \sin a x}\right)^2} = \frac 1 {2 a} \tan \left({\frac \pi 4 + \frac {a x} 2}\right) + \frac 1 {6 a} \tan^3 \left({\frac \pi 4 + \frac {a x} 2}\right) + C$
Converse Hinge Theorem
If two triangles have two pairs of sides which are the same length, the triangle in which the third side is longer also has the larger angle contained by the first two sides. {{:Euclid:Proposition/I/25}}
Definite Integral over Reals of Exponential of -(a x^2 plus b x plus c)
:$\displaystyle \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} }$
Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2)
:$\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$
Invariance of Extremal Length under Conformal Mappings
Let $X, Y$ be Riemann surfaces (usually, subsets of the complex plane). Let $\phi: X \to Y$ be a conformal isomorphism between $X$ and $Y$. Let $\Gamma$ be a family of rectifiable curves (or, more generally, of unions of rectifiable curves) in $X$. Let $\Gamma'$ be the family of their images under $\phi$. Then $\Gamma$ and $\Gamma'$ have the same extremal length: :$\map \lambda \Gamma = \map \lambda {\Gamma'}$
Set is Subset of Intersection of Supersets/Set of Sets
Let $T$ be a set. Let $\mathbb S$ be a set of sets. Suppose that for each $S \in \mathbb S$, $T \subseteq S$. Then: :$T \subseteq \displaystyle \bigcap \mathbb S$
Membership Relation is Not Symmetric
Let $\Bbb S$ be a set of sets in the context of pure set theory Let $\RR$ denote the membership relation on $\Bbb S$: :$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$ $\RR$ is not in general a symmetric relation.
Complementary Error Function of Zero
:$\displaystyle \map \erfc 0 = 1$
Primitive of Cosine of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\cos a x \rd x} x       | r = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C }} {{eqn | r = \ln \size x - \frac {\paren {a x}^2} {2 \times 2!} + \frac {\paren {a x}^4} {4 \times 4!} - \frac {\paren {a x}^6} {6 \times 6!} - \cdots + C }} {{end-eqn}}
Primitive of Cosine of a x over x squared
:$\displaystyle \int \frac {\cos a x \ \mathrm d x} {x^2} = \frac {-\cos a x} x - a \int \frac {\sin a x \ \mathrm d x} x$
Definite Integral to Infinity of x over Hyperbolic Sine of a x
:$\displaystyle \int_0^\infty \frac x {\sinh a x} \rd x = \frac {\pi^2} {4 a^2}$
Empty Set from Principle of Non-Contradiction
The empty set can be characterised as: :$\O := \set {x: x \in E \text { and } x \notin E}$ where $E$ is an arbitrary set.
Primitive of Reciprocal of Square of 1 plus Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\left({1 + \cos a x}\right)^2} = \frac 1 {2a} \tan \frac {a x} 2 + \frac 1 {6 a} \tan^3 \frac {a x} 2 + C$
Definite Integral from 0 to 2 Pi of Reciprocal of One minus 2 a Cosine x plus a Squared
:$\displaystyle \int_0^{2 \pi} \frac {\d x} {1 - 2 a \cos x + a^2} = \frac {2 \pi} {1 - a^2}$
Divisibility by 7
An integer $X$ with $n$ digits ($X_0$ in the ones place, $X_1$ in the tens place, and so on) is divisible by $7$ : :$\displaystyle \sum_{i \mathop = 0}^{n - 1} \paren {3^i X_i}$ is divisible by $7$.
Intersection is Empty and Union is Universe if Sets are Complementary
Let $A$ and $B$ be subsets of a universe $\Bbb U$. Then: :$A \cap B = \O$ and $A \cup B = \Bbb U$ : :$B = \relcomp {\Bbb U} A$ where $\relcomp {\Bbb U} A$ denotes the complement of $A$ with respect to $\Bbb U$.
Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x
:$\displaystyle \int_0^\infty \paren {\frac 1 {e^x - 1} - \frac {e^{-x} } x} \rd x = \gamma$
Set Intersection expressed as Intersection Complement
Let $A$ and $B$ be subsets of a universal set $\Bbb U$. Let $\uparrow$ denote the operation on $A$ and $B$ defined as: :$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$ where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$. Then: :$A \cap B = \paren {A \uparrow B} \uparrow \paren {A \uparrow B}$
Primitive of Power of Secant of a x by Tangent of a x
:$\displaystyle \int \sec^n a x \tan a x \ \mathrm d x = \frac {\sec^n a x} {n a} + C$
Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x
:$\displaystyle \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$
Singleton of Subset is Element of Powerset of Powerset
Let $S \subseteq T$ where $S$ and $T$ are both sets. Then: :$\set S \in \powerset {\powerset T}$ where $\powerset T$ denotes the power set of $T$.
Power Function is Completely Multiplicative
Let $K$ be a field. Let $z \in K$. Let $f_z: K \to K$ be the mapping defined as: :$\forall x \in K: f_z \left({x}\right) = x^z$ Then $f_z$ is completely multiplicative.
Primitive of Cosine of a x over p plus q of Sine of a x
:$\displaystyle \int \frac {\cos a x \rd x} {p + q \sin a x} = \frac 1 {a q} \ln \left\vert{p + q \sin a x}\right\vert + C$
Primitive of Cosine of a x over Power of p plus q of Sine of a x
:$\displaystyle \int \frac {\cos a x \ \mathrm d x} {\left({p + q \sin a x}\right)^n} = \frac {-1} {a q \left({n - 1}\right) \left({p + q \sin a x}\right)^{n - 1} } + C$
Number of Quadratic Residues of Prime
Let $p$ be an odd prime. Then $p$ has $\dfrac {p-1} 2$ quadratic residues and $\dfrac {p-1} 2$ quadratic non-residues. The quadratic residues are congruent modulo $p$ to the integers $1^2, 2^2, \ldots, \left({\dfrac {p-1} 2}\right)$.
Limit to Infinity of Fresnel Sine Integral Function
:$\displaystyle \lim_{x \mathop \to \infty} \map {\operatorname S} x = \frac 1 2$