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Catalog of special functions

From Citizendium, the Citizens' Compendium

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Algebraic functions

Complex parts

Elementary transcendental functions

Name	Notation
Exponential function	$exp(x)$ , $e x$
Natural logarithm	$log(x)$ , $ln(x)$

Trigonometric functions:

Name	Notation	Triangle formula	Exponential formula
Sine	$sin(x)$	Opposite / Hypotenuse	$(e i x - e - i x) / 2 i$
Cosine	$cos(x)$	Adjacent / Hypotenuse	$(e i x + e - i x) / 2$
Tangent	$tan(x)$	Opposite / Adjacent	$- i (e i x - e - i x) / (e i x + e - i x)$
Cosecant	$csc(x)$	Hypotenuse / Opposite
Secant	$sec(x)$	Hypotenuse / Adjacent
Cotangent	$cot(x)$	Adjacent / Opposite

Hyperbolic functions:

Name	Notation	Exponential formula
Hyperbolic sine	$sinh(x)$	$(e x - e - x) / 2$
Hyperbolic cosine	$cosh(x)$	$(e x + e - x) / 2$
Hyperbolic tangent	$tanh(x)$	$(e x - e - x) / (e x + e - x)$
Hyperbolic cosecant	$csch(x)$	$2 / (e x - e - x)$
Hyperbolic secant	$sech(x)$	$2 / (e x + e - x)$
Hyperbolic cotangent	$coth(x)$	$(e x + e - x) / (e x - e - x)$

Inverse trigonometric functions:

Name	Notation	Triangle formula	Exponential formula
Arcsine	$arcsin(x)$
Arccosine	$arccos(x)$
Arctangent	$arctan(x)$
Arccosecant	$arccsc(x)$
Arcsecant	$arcsec(x)$
Arccotangent	$arccot(x)$

Inverse hyperbolic functions:

Name	Notation	Logarithmic formula
Inverse hyperbolic sine	$arcsinh(x)$	$\ln{(x+\sqrt{x^2+1)}}$
Inverse hyperbolic cosine	$arccosh(x)$	$\ln{(x+\sqrt{x^2-1})}$
Inverse hyperbolic tangent	$arctanh(x)$	$\frac{1}{2}\ln{\frac{1+x}{1-x}}$
Inverse hyperbolic cosecant	$arccsch(x)$
Inverse hyperbolic secant	$arcsech(x)$
Inverse hyperbolic cotangent	$arccoth(x)$

Other:

Exponential integral related

Function	Notation	Definition
Exponential integral	$Ei(x)$	$\textstyle -\int_{-x}^{\infty} \frac{e^{-t}}{t} \, dt$
Logarithmic integral	$li(x)$	$\textstyle \int_0^x \frac{1}{\ln t} \, dt$

Trigonometric integrals:

Function	Notation	Definition
Sine integral	$Si(x)$	$\textstyle \int_0^x \frac{\sin t}{t} \, dt$
Hyperbolic sine integral	$Shi(x)$	$\textstyle \int_0^x \frac{\sinh t}{t} \, dt$
Cosine integral	$Ci(x)$	$\textstyle \gamma + \ln x + \int_0^x \frac{\cos t - 1}{t} \, dt$
Hyperbolic cosine integral	$Chi(x)$	$\textstyle \gamma + \ln x + \int_0^x \frac{\cosh t - 1}{t} \, dt$

Note: $γ$ is Euler's constant

Related to the normal distribution:

Name	Notation	Definition
Gaussian function	none standardized	$f(x) = a e^{-(x-b)^2/c^2}$
Error function	$erf(x)$	$\textstyle \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$
Complementary error function	$erfc(x)$	$1 - erf(x)$

See also gamma related functions below; in particular, the incomplete gamma functions.

Bessel function related

Elliptic integrals

Orthogonal polynomials

See catalog of orthogonal polynomials for a more detailed listing.

Name	Notation	Interval	Weight function	$f 0$ , $f 1$ , $f 2$ , $f 3$ , ...
Chebyshev (first kind)	$T n$	$- 1,1$	$(1 - x 2) - 1 / 2$	$1$ , $x$ , $2 x 2 - 1$ , $4 x 3 - 3 x$ , ...
Chebyshev (second kind)	$U n$	$- 1,1$	$(1 - x 2) 1 / 2$	$1$ , $2 x$ , $4 x 2 - 1$ , $8 x 3 - 4 x$ , ...
Legendre	$P n$	$- 1,1$	$1$	$1$ , $x$ , ${\textstyle \frac{1}{2}}$ $(3 x 2 - 1)$ , ${\textstyle \frac{1}{2}}$ $(5 x 3 - 3 x)$ , …
Hermite	$H n$	$-\infty,\infty$	$e^{-x^2}$
Laguerre	$L n$	$0,\infty$	$e - x$
Associated Laguerre	$L_n^{(\alpha)}$	$0,\infty$	$x α e - x$

Factorial and gamma related

Name	Notation	Discrete formula	Continuous formula
Factorial	$x!$	$1 \cdot 2 \cdot 3 \cdots x$	$Γ(x + 1)$
Gamma function	$Γ(x)$	$(x - 1)!$	$Γ(x)$
Double factorial	$x!!$	$1 \cdot 3 \cdot 5 \cdots x \;\;(x \; \mathrm{odd})$ $2 \cdot 4 \cdot 6 \cdots x \;\;(x \; \mathrm{even})$	$\frac{\Gamma(x+1)}{2^\frac{x-1}2 \Gamma(\frac{x+1}2)}\;\;(x \; \mathrm{odd})$ $2^\frac{x-1}2 \Gamma(\frac{x+1}2) \;\;(x \; \mathrm{even})$
Binomial coefficient	$n \choose k$	$\frac{n!}{k!(n-k)!}$	$\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}$
Rising factorial	$(x) (n)$	$\frac{(x+n-1)!}{(x-1)!}$	$\frac{\Gamma(x+n)}{\Gamma(x)}$
Falling factorial	$(x) (n)$	$\frac{x!}{(x-n)!}$	$\frac{\Gamma(x+1)}{\Gamma(x-n+1)}$
Beta function	$Β(x, y)$		$\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$
Harmonic number	$H n$	$\textstyle 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$	$γ + ψ(n + 1)$
Digamma function	$ψ(x),ψ (0) (x)$	$H x - 1 - γ$	$\begin{matrix}\frac{d}{dx}\end{matrix} \ln \Gamma(x)$
Polygamma function (of order m)	$ψ (m) (x)$		$\left(\begin{matrix}\frac{d}{dx}\end{matrix}\right)^{m+1} \ln \Gamma(x)$