Special function/Catalogs/Catalog

Algebraic functions

• Polynomials
  • Linear functions
  • Quadratic functions
  • Cubic functions
• Rational functions
• Power functions
  • Square root
  • Cube root

Complex parts

• Absolute value
• Real part
• Imaginary part
• Complex conjugate
• Complex argument
• Sign function

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	$(e^{ix}-e^{-ix})/2i$
Cosine	$\cos(x)$	Adjacent / Hypotenuse	$(e^{ix}+e^{-ix})/2$
Tangent	$\tan(x)$	Opposite / Adjacent
Cosecant	$\csc(x)$	Hypotenuse / Opposite
Secant	$\sec(x)$	Hypotenuse / Adjacent
Cotangent	$\cot(x)$	Adjacent / Opposite

Other

• Lambert W-function

Nonelementary integrals

• Exponential integral
• Logarithmic integral
• Trigonometric integrals
  • Sine integral
  • Cosine integral
  • Hyperbolic sine integral
  • Hyperbolic cosine integral

Bessel function related

• Airy functions
• Bessel functions
• Fresnel integrals

Orthogonal polynomials

See catalog of orthogonal polynomials for a more detailed listing.

Name	Notation	Interval	Weight function
Chebyshev (first kind)	$T_{n}$	$-1,1$	$(1-x^{2})^{-1/2}$
Chebyshev (second kind)	$U_{n}$	$-1,1$	$(1-x^{2})^{1/2}$
Legendre	$P_{n}$	$-1,1$	$1$
Hermite	$H_{n}$	$-\infty ,\infty$	$e^{-x^{2}}$
Laguerre	$L_{n}$	$0,\infty$	$e^{-x}$
Associated Laguerre	$L_{n}^{(\alpha )}$	$0,\infty$	$x^{\alpha }e^{-x}$

Factorial and gamma related

Name	Notation	Discrete formula	Continuous formula
Factorial	$x!$	$1\cdot 2\cdot 3\cdots x$	$\Gamma (x+1)$
Gamma function	$\Gamma (x)$	$(x-1)!$	$\Gamma (x)$
Double factorial	$x!!$	$1\cdot 3\cdot 5\cdots x\;\;(x\;\mathrm {odd} )$ $2\cdot 4\cdot 6\cdots x\;\;(x\;\mathrm {even} )$	${\begin{matrix}{\frac {1}{\sqrt {\pi }}}\end{matrix}}\;2^{(x+1)/2}\;\Gamma (x/2+1)$
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	$\mathrm {\mathrm {B} } (x,y)$		${\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}$
Harmonic number	$H_{n}$	$\textstyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}$	$\gamma +\psi (n+1)$
Digamma function	$\psi (x),\psi ^{(0)}(x)$	$H_{x-1}-\gamma$	${\begin{matrix}{\frac {d}{dx}}\end{matrix}}\log \Gamma (x)$
Polygamma function (of order m)	$\psi ^{(m)}(x)$		$\left({\begin{matrix}{\frac {d}{dx}}\end{matrix}}\right)^{m+1}\log \Gamma (x)$

• Incomplete gamma function
• Incomplete beta function
• Regularized gamma function
• Regularized beta function
• Barnes G-function

Notes:

• $\gamma$ is Euler's constant
• The polygamma functions are generalized to continuous m by the Hurwitz zeta function

Zeta function related

• Riemann zeta function
• Hurwitz zeta function
• Polylogarithm

Hypergeometric functions

Note: many of the preceding functions are special cases of the following:

• Hypergeometric functions
• Meijer G-function

See also

• Catalog of mathematical constants
• Catalog of probability distributions