Revision as of 05:29, 16 May 2015

Special functions are ubiquitous in mathematics and applications of mathematics. The term applies to such a wide range of functions that no single source contains all of them. We aim to remedy this problem.

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Special functions

Airy Ai	Barnes $G$	Beta	Binomial	Dedekind eta
Devil's staircase	Digamma	Dirichlet eta	Dirichlet beta	Error function	Exponential
Exponential integral	Factorial	Gamma	Hadamard gamma	Hyperfactorial	Hypergeometric ${}_pF_q$
Incomplete gamma	Klein invariant $J$	Log integral	Logarithm	Thomae	Polygamma
Polylogarithm	Partition	Reciprocal gamma	Lambert $W$	Minkowski $?$	Planck radiation
Pochhammer	Ramanujan $\tau$	Riemann-Siegel $Z$	Riemann $\xi$

Airy Bi
Automorphic forms and functions
Blancmange function
Carlitz-Goss Gamma function
Clausen functions
Dawson function
Eisenstein series
Heaviside step function
Hermite function
Jacobi theta functions
K-function
Legendre chi
Legendre function
Lerch transcendent
Mathieu function
Maass forms
Meijer G-function
Modular form
Mock modular forms
Mock theta functions
$p$-adic $L$ function
Painlevé transcendents
Riemann theta function
Sievert integral
Signum
Struve functions
Theta functions

Bessel functions

Bessel $J_{\nu}$

Bessel $Y_{\nu}$
Hankel $H_{\nu}^{(1)}$
Hankel $H_{\nu}^{(2)}$
Modified Bessel $I_{\nu}$
Modified Bessel $K_{\nu}$
Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$
Spherical Bessel $j_{\nu}$
Spherical Bessel $y_{\nu}$
Spherical Hankel $h_{\nu}^{(1)}$
Spherical Hankel $h_{\nu}^{(2)}$

Elliptic functions

Elliptic $E$	Elliptic $K$	Elliptic $\theta$	Jacobi $\mathrm{cn}$	Jacobi $\mathrm{cd}$	Jacobi $\mathrm{cs}$	Jacobi $\mathrm{dc}$
Jacobi $\mathrm{dn}$	Jacobi $\mathrm{ds}$	Jacobi $\mathrm{nc}$	Jacobi $\mathrm{nd}$	Jacobi $\mathrm{ns}$	Jacobi $\mathrm{sc}$
Jacobi $\mathrm{sd}$	Jacobi $\mathrm{sn}$	Modular $\lambda$	Weierstrass $P$

Jacobi elliptic functions
Weierstrass $\sigma$

Polynomials

Abel $p$	Associated Laguerre L	Bateman $Z$	Bernoulli $B$	Bessel poly	Chebyshev U
Chebyshev T	Dickson	Euler $E$	Fibonacci	Gegenbauer $C$	Hermite (phys)
Hermite (prob)	Jacobi $P$	Laguerre $L$	Legendre $P$	Lucas

Bell polynomial
Charlier polynomial
$q$-Hermite polynomial
Krawtchouk polynomial
Lagrange polynomial
Lommel polynomial
Mahler polynomial
Mott polynomial
Pidduck polynomial
Spread polynomial
Touchard polynomial
Rook polynomial

Trigonometric functions

Böhmer integral

Sinc

Standard trigonometric functions

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent

Inverse trigonometric functions

Arcsin

Arccos

Arctan

Arccsc

Arcsec

Arccot

Hyperbolic trigonometric functions

Sinh

Cosh

Tanh

Csch

Sech

Coth

Inverse hyperbolic trigonometric functions

Arcsinh

Arccosh

Arctanh

Arccsch

Arcsech

Arccoth

$*$-integral functions

Cosine integral

Cosh integral

Fresnel $C$

Exponential integral

Fresnel S

Sine integral

Sinh integral

Special functions in number theory

Euler totient	Mangoldt	Mertens	Möbius	Liouville $\lambda$	Prime counting
Riemann zeta	Sum of divisors

Chebyshev function
Dirichlet $L$-function
Greatest prime factor
Ramanujan's sum
Riemann prime counting

Zeta functions

A directory of zeta functions.

Hurwitz zeta

Jacobi zeta

Prime zeta $P$

Riemann zeta

Weierstrass zeta

Airy zeta function
Arithmetic zeta function
Arakawa-Kaneko zeta function
Arithmetic zeta function
Artin-Mazur zeta function
Barnes zeta function
Beurling zeta function
Cotangent zeta function
Dedekind zeta function
Epstein zeta function
Goss zeta function
Hasse-Weil zeta function
Height zeta function
Igusa zeta function
Ihara zeta function
Lefschetz zeta function
Lerch zeta function
Local zeta function
Matsumoto zeta function
Minakshisudaram-Pleijel zeta function
Motivic zeta function
Multiple zeta function
$p$-adic zeta function
Reciprocal zeta function
Ruelle zeta function
Secant zeta function
Selberg zeta function
Shimizu zeta function
Shintani zeta function
Witten zeta function

$q$-Calculus

$q$-Binomial

$q$-Gamma

$q$-factorial

Euler $\phi$

Basic Fourier series
Basic hypergeometric series $\phi$
Basic hypergeometric series $\psi$
Continuous $q$-Hermite polynomials
Hahn-Exton $q$-Bessel
Gosper $q$-sine
Jackson $q$-Bessel (1)
Jackson $q$-Bessel (2)
Jackson $q$-Bessel (3)
$q$-analog
$q$-Bernoulli numbers
$q$-Bessel function
$q$-Beta function
$q$-$\cos$
$q$-$\mathrm{Cos}$
$q$-derivative
$q$-exponential e
$q$-exponential E
$q$-Fibonacci polynomials
$q$-Gaussian distribution
$q$-Hermite polynomial
$q$-Hypergeometric function
$q$-numbers
$q$-Pochhammer symbol
$q$-Polygamma function
$q$-shifted factorial
$q$-$\sin$
$q$-$\mathrm{Sin}$
$q$-Theta function
Tsallis $q$-exponential

Special sequences

Bell numbers
Bernoulli numbers ($B_0 = 1, B_1 = \pm \frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = −\frac{1}{30}, B_5 = 0, B_6 = \frac{1}{42}, B_7 = 0, B_8 = −\frac{1}{30}$)
Euler numbers
Genocchi numbers
Harmonic number
Lah numbers
Mersenne numbers
Stirling numbers of the first kind
Stirling numbers of the second kind

Figurate numbers

Square number
Triangular numbers
Tetrahedral numbers

Special constants

Apéry's constant ($\zeta(3)=1.2020569031595942854 \ldots$)
Catalan constant ($0.915965594177219015054603514932384110774 \ldots$)
Chaitin's constant
Champernowne constant ($0.12345678910111213141516171819202122232425\ldots$)
Conway's constant ($1.303577269034 \ldots$)
Copeland-Erdős constant ($0.23571113171923...$)
Euler's number ($e=2.71828182846 \ldots$)
Erdős-Borwein Constant ($E=1.606695152415291763\ldots$)
Euler-Mascheroni constant ($\gamma=0.5772156649015328606065120900824024310421593359399235\ldots$)
Feigenbaum constants ($4.669201609102990671853203820466201617258185577475768632745651 \ldots$)
Fransén–Robinson constant ($2.807770242028519365221501186557772932308085920930198291220054 \ldots$)
Gauss' constant
Gelfond constant ($e^{\pi}=23.14069263277926900572908636794854738026610624260021199344504 \ldots$)
Gelfond–Schneider constant ($2^{\sqrt{2}}=2.665144142690225188650297249873139848274211313714659492835979 \ldots$)
Glaisher–Kinkelin constant ($A = 1.2824271291 \ldots$)
Golden ratio ($\phi=1.618033988749894848204586834365638117720309179805762862135448 \ldots$)
Imaginary number ($i=\sqrt{-1}$)
Kaprekar's constant
Khinchin's constant ($K=2.685452001065306445309714835481795693820382293994462953051152 \ldots$)
Legendre's constant ($B=1$)
Lefschetz number
Lévy's constant
Mills' constant ($M=1.306377883863080690468614492602605712916784585156713644368053 \ldots$)
Pi ($\pi=3.141592653589793238462643383279502884197169399375105820974944 \ldots$)
$q$-Bernoulli numbers
$\pi_q$
Silver ratio ($\delta_s=2.4142135623730950488 \ldots$)
Soldner's Constant ($\mu = 1.45136923488338105028396848589202744949\ldots$)

Numbers

Algebraic number $\overline{\mathbb{Q}}$
Complex number $\mathbb{C}$
Integer $\mathbb{Z}$
Irrational number $\mathbb{R} \setminus \mathbb{Q}$
Natural number $\mathbb{N}$
Prime number
Rational number $\mathbb{Q}$
Real number $\mathbb{R}$

Tools from calculus

Absolute maximum
Absolute minimum
Analytic continuation
Continuous
Derivative
Difference equation of hypergeometric type
Integration by parts
Laplace transform
Lebesgue integral
Local maximum
Local minimum
Ratio test
Riemann integral
Rolle's theorem
Taylor series
Weierstrass factorization

Other

Solution manual to Rainville's "Special Functions"
"Special Functions" by Leon Hall
Relations between special functions by John D. Cook
NIST Digital Library of Mathematical Functions
Mathematical Constants by Steven R. Finch
T.H Koornwinder's list of resources
Victor Moll's proofs of Gradshteyn and Ryzhik integrals
Abramowitz and Stegun
LMFDB, the database of L-functions, modular forms, and related objects.
Phillipe Flajolet's /Analytic Combinatorcs/
Gallery of mathematical functions
Online number theory lecture notes and teaching materials
The On-Line Encyclopedia of Integer Sequences