hcju/proofwikips huggingface dataset:
Series Expansion of Bessel Function of the First Kind/Negative Index
{{begin-eqn}} {{eqn | l = \map {J_{-n} } x       | r = \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}       | c =  }} {{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {k + 1 - n} } \paren {\dfrac x 2}^{2 k - n}       | c =  }} {{end-eqn}}
Bessel Function of the First Kind for Imaginary Argument
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map {J_n } {i x} = i^{-n} \, \map {I_n} x$ where: :$i$ denotes the imaginary unit :$\map {I_n} x$ denotes the modified Bessel function of the first kind of order $n$.
Integral to Infinity of Shifted Dirac Delta Function by Continuous Function
Let $\map \delta x$ denote the Dirac delta function. Let $g$ be a continuous real function. Let $a \in \R_{\ge 0}$ be a positive real number. Then: :$\displaystyle \int_0^{+ \infty} \map \delta {x - a} \, \map g x \rd x = \map g a$
Function which is Zero except on Countable Set of Points is Null
Let $S \subseteq \R$ be a subset of $\R$ such that $S$ is countable, either finite or countably infinite. Let $f: \R \to \R$ be a real function such that: :$\forall x \in \R \setminus S: \map f x = 0$ That is, except perhaps for the elements of $S$, the value of $f$ is zero. Then $f$ is a null function.
Laplace Transform of Bessel Function of the First Kind of Order Zero
Let $J_0$ denote the Bessel function of the first kind of order $0$. Then the Laplace transform of $J_0$ is given as: :$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$
Laplace Transform of Bessel Function of the First Kind
Let $J_n$ denote the Bessel function of the first kind of order $n$. Then the Laplace transform of $J_n$ is given as: :$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$
Laplace Transform of Sine of Root
:$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$ where $\laptrans f$ denotes the Laplace transform of the function $f$.
Laplace Transform of Cosine of Root over Root
:$\laptrans {\dfrac {\cos \sqrt t} {\sqrt t} } = \sqrt {\dfrac \pi s} \, \map \exp {-\dfrac 1 {4 s} }$ where $\laptrans f$ denotes the Laplace transform of the function $f$.
Laplace Transform of Error Function of Root
:$\laptrans {\map \erf {\sqrt t} } = \dfrac 1 {s \sqrt {s + 1} }$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\erf$ denotes the error function
Laplace Transform of Sine Integral Function
:$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Si$ denotes the sine integral function
Laplace Transform of Cosine Integral Function
:$\laptrans {\map \Ci t} = \dfrac {\map \ln {s^2 + 1} } {2 s}$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Ci$ denotes the cosine integral function.
Laplace Transform of Heaviside Step Function
Let $\map {u_c} t$ denote the Heaviside step function: :$\map {u_c} t = \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$ The Laplace transform of $\map {u_c} t$ is given by: :$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} } s$ for $\map \Re s > c$.
Laplace Transform of Dirac Delta Function
Let $\map \delta t$ denote the Dirac delta function. The Laplace transform of $\map \delta t$ is given by: :$\laptrans {\map \delta t} = 1$
Laplace Transform of Null Function
Let $\mathcal N: \R \to \R$ be a null function. The Laplace transform of $\map {\mathcal N} t$ is given by: :$\laptrans {\map {\mathcal N} t} = 0$
Laplace Transform of Function of t minus a/Proof 1
Let $f$ be a function such that $\laptrans f$ exists. Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$. Let $a \in \C$ or $\R$ be constant. {{:Laplace Transform of Function of t minus a}}
Laplace Transform of Function of t minus a/Proof 2
Let $f$ be a function such that $\laptrans f$ exists. Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$. Let $a \in \C$ or $\R$ be constant. {{:Laplace Transform of Function of t minus a}}
Laplace Transform of Sine of t over t
Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$
Integral to Infinity of Function over Argument
:$\displaystyle \int_0^\infty {\dfrac {\map f t} t} = \int_0^{\to \infty} \map F u \rd u$ provided the integrals converge.
Integral to Infinity of Exponential of -t^2
:$\displaystyle \int_0^\infty \map \exp {-t^2} \rd t = \dfrac {\sqrt \pi} 2$
Laplace Transform of Real Power
Let $n$ be a constant real number such that $n > -1$ Let $f: \R \to \R$ be the real function defined as: :$\map f t = t^n$ Then $f$ has a Laplace transform given by: {{begin-eqn}} {{eqn	| l = \laptrans {\map f t} 	| r = \int_0^\infty e^{-s t} t^n \rd t }} {{eqn	| r = \frac {\map \Gamma {n + 1} } {s^{n + 1} } }} {{end-eqn}} where $\Gamma$ denotes the gamma function.
Laplace Transform of Reciprocal of Square Root
:$\laptrans {\dfrac 1 {\sqrt t} } = \sqrt {\dfrac \pi s}$ where $\laptrans f$ denotes the Laplace transform of the real function $f$.
Gamma Function of 3 over 2
:$\map \Gamma {\dfrac 3 2} = \dfrac {\sqrt \pi} 2$
Gamma Function of Minus 3 over 2
:$\map \Gamma {-\dfrac 3 2} = \dfrac {4 \sqrt \pi} 3$
Gamma Function of Zero
:$\map \Gamma 0$ is not defined.
Gamma Function of Minus 1
:$\map \Gamma {-1}$ is not defined.
Gamma Function of Minus 2
:$\map \Gamma {-2}$ is not defined.
Integral to Infinity of Bessel Function of First Kind order Zero
:$\displaystyle \int_0^\infty \map {J_0} t \rd t = 1$ where $J_0$ denotes theBessel function of the first kind of order $0$.
Integral to Infinity of e^-t by Error Function of Root t
:$\displaystyle \int_0^\infty e^{-t} \erf \sqrt t \rd t = \dfrac {\sqrt 2} 2$
Second Derivative of Laplace Transform
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, twice differentiable on any closed interval $\closedint 0 a$. Let $\laptrans f = F$ denote the Laplace transform of $f$. Then, everywhere that $\dfrac {\d^2} {\d s^2} \laptrans f$ exists: :$\dfrac {\d^2} {\d s^2} \laptrans {\map f t} = \laptrans {t^2 \, \map f t}$
Laplace Transform of Multiple Integral
:$\displaystyle \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$ wherever $\laptrans f$ exists.
Integral of Reciprocal is Divergent/Unbounded Above
:$\displaystyle \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$
Integral of Reciprocal is Divergent/To Zero
:$\displaystyle \int_\gamma^1 \frac {\d x} x \to -\infty$ as $\gamma \to 0^+$
Convergence of P-Series/Lemma
Let $p = x + i y$ be a complex number where $x, y \in \R$ such that: :$x > 0$ :$x \ne 1$ Then: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^x}$ converges  $\displaystyle \lim_{P \mathop \to \infty} \dfrac {P^{1 - x} } {1 - x}$ converges.
Integral to Infinity of Reciprocal of Power of x
The improper integral :$\displaystyle \int_1^\infty \dfrac {\d t} {t^x}$ exists  $x > 1$.
Convergence of P-Series/Real
Let $p \in \R$ be a real number. Then the $p$-series: :$\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$ is convergent  $p > 1$.
Real Number Ordering is Transitive
Let $a, b, c \in \R$ such that $a > b$ and $b > c$. Then: :$a > c$
Real Number Ordering is Compatible with Multiplication/Positive Factor
:$\forall a, b, c \in \R: a < b \land c > 0 \implies a c < b c$
Real Number Ordering is Compatible with Multiplication/Negative Factor
:$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$
Rational Power of Product of Real Numbers
Let $r, s \in \R_{> 0}$ be (strictly) positive real numbers.  Let $x \in \Q$ be a rational number. Let $r^x$ be defined as $r$ to the power of $x$. Then: :$\paren {r s}^x = r^x s^x$
Sign of Quadratic Function Between Roots
Let $a \in \R_{>0}$ be a (strictly) positive real number. Let $\alpha$ and $\beta$, where $\alpha < \beta$, be the roots of the quadratic function: :$\map Q x = a x^2 + b x + c$ whose discriminant $b^2 - 4 a c$ is (strictly) positive. Then: :$\begin {cases} \map Q x < 0 & : \text {when $\alpha < x < \beta$} \\ \map Q x > 0 & : \text {when $x < \alpha$ or $x > \beta$} \end {cases}$
Minimum Value of Real Quadratic Function
Let $a \in \R_{>0}$ be a (strictly) positive real number. Consider the quadratic function: :$\map Q x = a x^2 + b x + c$ $\map Q x$ achieves a minimum at $x = -\dfrac b {2 a}$, at which point $\map Q x = c - \dfrac {b^2} {4 a}$.
Number of Type Rational r plus s Root 2 is Irrational
Let $r, s \in \Q$ be rational numbers. Then $r + s \sqrt 2$ is irrational.
Roots of Quadratic with Rational Coefficients of form r plus s Root 2
Consider the quadratic equation: :$(1): \quad a^2 x + b x + c = 0$ where $a, b, c$ are rational. Let $\alpha = r + s \sqrt 2$ be one of the roots of $(1)$. Then $\beta = r - s \sqrt 2$ is the other root of $(1)$.
Supremum is not necessarily Greatest Element
Let $\struct {S, \preceq}$ be an ordered set. Let $T$ admit a supremum in $S$. Then the supremum of $T$ in $S$ is not necessarily the greatest element of $T$.
Closed Interval Defined by Absolute Value
:$\set {x \in \R: \size {\xi - x} \le \delta} = \closedint {\xi - \delta} {\xi + \delta}$ where $\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$.
Open Interval Defined by Absolute Value
:$\set {x \in \R: \size {\xi - x} < \delta} = \openint {\xi - \delta} {\xi + \delta}$ where $\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$.
Distance from Subset of Real Numbers to Element
:$x \in S \implies \map d {x, S} = 0$
Distance from Subset of Real Numbers to Supremum
Let $S$ be bounded above such that $\xi = \sup S$. Then: :$\map d {\xi, S} = 0$
Distance from Subset of Real Numbers to Infimum
Let $S$ be bounded below such that $\xi = \inf S$. Then: :$\map d {\xi, S} = 0$
Real Number at Distance Zero from Closed Real Interval is In Interval
Let $I \subseteq \R$ be a closed real interval. Then: :$\map d {x, I} = 0 \implies x \in I$
Existence of Real Number at Distance Zero from Open Real Interval not in Interval
Let $I \subseteq \R$ be an open real interval such that $I \ne \O$ and $I \ne \R$. Then: :$\exists x \notin I: \map d {x, I} = 0$
Distance from Subset of Real Numbers to Element/Proof 1
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. Then: {{:Distance from Subset of Real Numbers to Element}}
Distance from Subset of Real Numbers to Element/Proof 2
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. Then: {{:Distance from Subset of Real Numbers to Element}}
Distance from Subset of Real Numbers to Supremum/Proof 1
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Supremum}}
Distance from Subset of Real Numbers to Supremum/Proof 2
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Supremum}}
Distance from Subset of Real Numbers to Infimum/Proof 1
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Infimum}}
Distance from Subset of Real Numbers to Infimum/Proof 2
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Infimum}}
Infimum of Set of Reciprocals of Positive Integers
Let $S$ be the subset of the set of real numbers defined as: :$S = \set {\dfrac 1 n: n \in \Z_{>0} }$ Then: :$\inf S = 0$ where $\inf S$ denotes the infimum of $S$.
Set of Numbers of form n - 1 over n is Bounded Above
Let $S$ be the subset of the set of real numbers $\R$ defined as: :$S = \set {\dfrac {n - 1} n: n \in \Z_{>0} }$ $S$ is bounded above with supremum $1$. $S$ has no greatest element.
Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element
Let $S \subseteq \Q$ be the subset of the set of rational numbers defined as: :$S = \set {r \in \Q: 0 < r < 1}$ Then $S$ has no greatest or smallest element. However, $S$ has a supremum $1$ and an infimum $0$.
Between two Real Numbers exists Irrational Number
Let $a, b \in \R$ be real numbers where $a < b$.  Then there exists an irrational number $\xi \in \R \setminus \Q$ such that: :$a < \xi < b$
Sequence of Powers of Reciprocals is Null Sequence/Real Index
Let $r \in \R_{>0}$ be a strictly positive real number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as: : $x_n = \dfrac 1 {n^r}$ Then $\sequence {x_n}$ is a null sequence.
Odd Order Derivative of Even Function Vanishes at Zero
Let $X$ be a symmetric subset of $\R$ containing $0$.  Let $n$ be a positive integer.  Let $f:X \to \R$ be an even function.  Let $f$ be at least $\paren{2 n + 1}$-times differentiable. Then: :$\map {f^{\paren {2 n + 1} } } 0 = 0$
Reciprocal of Null Sequence/Corollary
:$x_n \to \infty$ as $n \to \infty$  $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$
Index of Subsequence not Less than its Index
Let $\sequence {x_n}_{n \mathop \ge 1}$ be a sequence in a set $S$. Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\forall n \in \N_{>0}: n_r \ge r$
Even Order Derivative of Odd Function Vanishes at Zero
Let $X$ be a symmetric subset of $\R$ containing $0$.  Let $n$ be a positive integer.  Let $f:X \to \R$ be an odd function.  Let $f$ be at least $\paren {2 n}$-times differentiable. Then:  :$\map {f^{\paren {2 n} } } 0 = 0$
Geometric Mean of Reciprocals is Reciprocal of Geometric Mean
Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers. Let $G_n$ denote the geometric mean of $x_1, x_2, \ldots, x_n$. Let ${G_n}'$ denote the geometric mean of their reciprocals $\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 {x_n}$. Then: :${G_n}' = \dfrac 1 {G_n}$
Geometric Mean is Never Less than Harmonic Mean
Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers. Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$. Then $G_n \ge H_n$.
Harmonic Mean of two Real Numbers is Between them
Let $a, b \in \R_{\ne 0}$ be non-zero real numbers such that $a < b$. Let $\map H {a, b}$ denote the narmonic mean of $a$ and $b$. Then: :$a < \map H {a, b} < b$
Arithmetic Mean of two Real Numbers is Between them
Let $a, b \in \R_{\ne 0}$ be non-zero real numbers such that $a < b$. Let $\map A {a, b}$ denote the narmonic mean of $a$ and $b$. Then: :$a < \map A {a, b} < b$
Raw Moment of Bernoulli Distribution
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:  :$\expect {X^n} = p$
Equivalence of Definitions of Local Basis
Let $T = \struct {S, \tau}$ be a topological space. Let $x$ be an element of $S$. {{TFAE|def = Local Basis}}
Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets
Let $\BB$ be a set of open neighborhoods of $x$ such that: :$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$
Equivalence of Definitions of Local Basis/Neighborhood Basis of Open Sets Implies Local Basis for Open Sets
Let $\BB$ be a set of open neighborhoods of $x$ such that: :every neighborhood of $x$ contains a set in $\BB$.
Arcsine of Zero is Zero
:$\arcsin 0 = 0$ where $\arcsin$ is the arcsine function.
Arcsine of One is Half Pi
:$\arcsin 1 = \dfrac \pi 2$ where $\arcsin$ is the arcsine function.
Real Sequence (1 + x over n)^n is Convergent
The sequence $\sequence {s_n}$ defined as: :$s_n = \paren {1 + \dfrac x n}^n$ is convergent.
Raw Moment of Poisson Distribution
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:  :$\displaystyle \expect {X^n} = \sum_{k \mathop = 0}^n \lambda^k {n \brace k}$ where $\displaystyle {n \brace k}$ is a Stirling number of the second kind.
Raw Moment of Exponential Distribution
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:  :$\expect {X^n} = n! \beta^n$
Central Moment of Exponential Distribution
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Let $n$ be a strictly positive integer.  Then the $n$th central moment $\mu_n$ of $X$ is given by:  :$\displaystyle \mu_n = n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}$
Product of Limits of Real Sequences (1 + x over n)^n and (1 - x over n)^n equals 1
Let $\sequence {a_n}$ be the sequence defined as: :$a_n = \paren {1 + \dfrac x n}^n$ Let $\sequence {b_n}$ be the sequence defined as: :$b_n = \paren {1 - \dfrac x n}^n$ Then the product of the limits of $\sequence {a_n}$ and $\sequence {b_n}$ equals $1$
Farey Sequence is not Convergent
Consider the Farey sequence: :$F = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$ $F$ is not convergent.
Skewness of Exponential Distribution
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Then the skewness $\gamma_1$ of $X$ is equal to $2$.
Excess Kurtosis of Geometric Distribution
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = 6 + \dfrac {p^2} {1 - p}$
Excess Kurtosis of Poisson Distribution
Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac 1 \lambda$
Expectation of Chi-Squared Distribution
Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the expectation of $X$ is given by:  :$\expect X = n$
Variance of Chi-Squared Distribution
Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the variance of $X$ is given by:  :$\var X = 2 n$
Skewness of Chi-Squared Distribution
Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \sqrt{\dfrac 8 n}$
Raw Moment of Chi-Squared Distribution
Let $n$ and $m$ be strictly positive integers.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the $m$th raw moment $\expect {X^m}$ of $X$ is given by:  :$\displaystyle \expect {X^m} = \prod_{k \mathop = 0}^{m - 1} \paren {n + 2 k}$
Sequence of Square Roots of Natural Numbers is not Cauchy
Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be the sequence in $\R$ defined as: :$x_n = \sqrt n$ Then, despite the fact that from Difference Between Adjacent Square Roots Converges: :$\size {\sqrt {n + 1} - \sqrt n} \to 0$ as $n \to \infty$ it is not the case that $\sequence {x_n}$ is a Cauchy sequence.
Square of Standard Gaussian Random Variable has Chi-Squared Distribution
Let $X \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the standard Gaussian distribution.  Then $X^2 \sim \chi^2_1$ where $\chi^2_1$ is the chi-square distribution with $1$ degree of freedom.
Sum of Chi-Squared Random Variables
Let $n_1, n_2, \ldots, n_k$ be strictly positive integers which sum to $N$. Let $X_i \sim \chi^2_{n_i}$ for $1 \le i \le k$, where $\chi^2_{n_i}$ is the chi-squared distribution with $n_i$ degrees of freedom.  Then:  :$\displaystyle X = \sum_{i \mathop = 1}^k X_i \sim \chi^2_N$
Moment Generating Function of Chi-Squared Distribution
Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the moment generating function of $X$, $M_X$, is given by:  :$\displaystyle \map {M_X} t = \begin{cases} \paren {1 - 2 t}^{-n / 2} & t < \frac 1 2 \\ \text{does not exist} & t \ge \frac 1 2\end{cases}$
Moment Generating Function of Linear Combination of Independent Random Variables
Let $X_1, X_2, \ldots, X_n$ be independent random variables.  Let $k_1, k_2, \ldots, k_n$ be real numbers. Let:  :$\displaystyle X = \sum_{i \mathop = 1}^n k_i X_i$ Let $M_{X_i}$ be the moment generating function of $X_i$ for $1 \le i \le n$. Then:  :$\displaystyle \map {M_X} t = \prod_{i \mathop = 1}^n \map {M_{X_i}} {k_i t}$ for all $t$ such that $M_{X_i}$ exists for all $1 \le i \le n$.
Variance of Random Sample from Gaussian Distribution has Chi-Squared Distribution
Let $X_1, X_2, \ldots, X_n$ form a random sample of size $n$ from the Gaussian distribution $\Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$.  Let: :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ and: :$\displaystyle s^2 = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$ Then:  :$\dfrac {\paren {n - 1} s^2} {\sigma^2} \sim \chi^2_{n - 1}$ where $\chi^2_{n - 1}$ is the chi-squared distribution with $n - 1$ degrees of freedom.
Expectation of F-Distribution
Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then the expectation of $X$ is given by: :$\expect X = \dfrac m {m - 2}$ for $m > 2$, and does not exist otherwise.
Variance of F-Distribution
Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then the variance of $X$ is given by:  :$\var X = \dfrac {2 m^2 \paren {m + n - 2} } {n \paren {m - 4} \paren {m - 2}^2}$ for $m > 4$, and does not exist otherwise.
Differential Equations for Shortest Path on 3d Sphere/Cartesian Coordinates
Let $M$ be a $3$-dimensional Euclidean space. Let $S$ be a sphere embedded in $M$. Let $\gamma$ be a curve on $S$. Let the chosen coordinate system be Cartesian. Let $\gamma$ begin at $\paren {x_0, y_0, z_0}$ and terminate at $\paren {x_1, y_1, z_1}$. Let $\map y x$, $\map z x$ be real functions. Let $\gamma$ connecting both endpoints be of minimum length. Then $\gamma$ satisfies the following equations of motion: :$2 y \map \lambda x - \dfrac \d {\d x} \dfrac {y'} {\sqrt {1 + y'^2 + z'^2} } = 0$ :$2 z \map \lambda x - \dfrac \d {\d x} \dfrac {z'} {\sqrt {1 + y'^2 + z'^2} } = 0$
Skewness of F-Distribution
Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {2 \paren {m + 2 n - 2} } {m - 6} \sqrt {\dfrac {2 \paren {m - 4} } {n \paren {m + n - 2} } }$ for $m > 6$, and does not exist otherwise.
Expectation of Chi Distribution
Let $n$ be a strictly positive integer.  Let $X \sim \chi_n$ where $\chi_n$ is the chi distribution with $n$ degrees of freedom.  Then the expectation of $X$ is given by:  :$\expect X = \sqrt 2 \dfrac {\map \Gamma {\paren {n + 1} / 2} } {\map \Gamma {n / 2} }$ where $\Gamma$ is the gamma function.