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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

Carlitz-Goss Gamma function
Clausen cosine
Clausen sine
Debye function
Dirichlet function
Distance to integers
Eisenstein series
Heaviside step function
Hermite function
Legendre chi
Legendre function
Lerch transcendent
Log barrier
MacRobert E
Maass forms
Meijer G-function
Modular form
Mock modular forms
Mock theta functions
Painlevé transcendents
Ramanujan theta function
Riemann-Landau xi
Riemann Siegel theta function
Riemann theta function
Sievert integral
Singular function
Theta functions
Unit step function
Voight function
Volterra function
Weierstrass elementary factors

Bessel functions and friends

Bessel functions
Airy functions
Hankel functions
Kelvin functions

Modified Struve function
Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$
Weber function

*-c functions

Normalized sinc

$*$-c functions

Continuous nowhere-differentiable functions and friends

Continuous nowhere differentiable functions

Besicovitch functions
Bolzano function
Cellérier function
Katsuura function
Keisswetter function
Koch curve
Knopp function
Lynch function
McCarthy function
Orlicz functions
Peano function
Petr function
Schoenberg function
Schwarz function
Sierpiński curve
Weierstrass nowhere differentiable function
Wen function

Defined by a differential equation

Buchstab function
Dickman–de Bruijn function
Mathieu function

Elliptic functions

Elliptic functions (general overview and definitions)

Jacobi Elliptic Functions

Incomplete Elliptic E
Incomplete Elliptic K
Jacobi elliptic functions
Weierstrass $\sigma$

Error function and friends

Dawson D-
Faddeeva function
Inverse complementary error function

Error functions

Figurate numbers

Tetrahedral numbers
Pyramidal numbers
Cubic numbers
Octahedral numbers

Polygonal numbers

Enneadecagonal numbers
Heptadecagonal numbers
Hexadecagonal numbers
Icosagonal numbers
Icosidigonal numbers
Icositetragonal numbers
Icositrigonal numbers
Icosihenagonal numbers
Myriagonal numbers
Octadecagonal numbers
Rectangular numbers

Gamma function and friends

Falling factorial
Incomplete beta function
Lower incomplete gamma
Multiple gamma function
Rising factorial

Hypergeometric series

Hypergeometric functions

Hypergeometric 0F3
Hypergeometric 1F1
Hypergeometric 1F2
Hypergeometric 2F0
Hypergeometric 2F3
Hypergeometric 3F2
Hypergeometric 4F1
Lauricella–Saran functions

*-integral functions

$\ast$-integral functions

Böhmer C
Böhmer S
Entire exponential integral

Jacobi theta functions

Jacobi theta functions

Logarithm function and friends

Logarithm and friends

Logarithm base a
Logarithm (multivalued)

Matrix functions

Matrix arcsin
Matrix exponential
Matrix cosh
Matrix sinh
Matrix logarithm
Matrix sine
Matrix cosine

Matrix hypergeometric functions

Matrix hypergeometric pFq
Matrix hypergeometric 0F3
Matrix hypergeometric 1F1
Matrix hypergeometric 1F2
Matrix hypergeometric 2F0
Matrix hypergeometric 2F3
Matrix hypergeometric 3F2
Matrix hypergeometric 4F1


Alexander operator
Bernardi operator
Fourier transform
Laplace transform
Libera operator
Mellin transform
Radon transform
Two-dimensional Laplace transform


Bernstein B

Probability distributions

Continuous distributions

Probability density funcitons

Arcsin pdf
Beta pdf
Cauchy pdf
Chi-squared pdf
Exponential pdf
F pdf
Gamma pdf
Laplace pdf
Log-normal pdf
Normal pdf
Pareto pdf
Student's t pdf
Continuous uniform pdf
Weibull pdf

Cumulative Density Functions

Arcsin cdf
Beta cdf
Cauchy cdf
Chi-squared cdf
Exponential cdf
F cdf
Gamma cdf
Laplace cdf
Log-normal cdf
Normal cdf
Pareto cdf
Student's t cdf
Continuous uniform cdf
Weibull cdf

Discrete distributions

Bernoulli distribution
Binomial distribution
Discrete uniform distribution
Geometric distribution
Hypergeometric distribution
Negative binomial distribution
Poisson distribution

Orthogonal polynomials

Adomian polynomials
Angelescu polynomials
Bell polynomial
Boole polynomials
Charlier polynomial
Cyclotomic polynomials
Denisyuk polynomials
Hahn polynomial
Humbert polynomials
$q$-Hermite polynomial
Krawtchouk polynomial
Lagrange polynomial
Lidstone polynomial
Lommel polynomial
Macdonald polynomials
Mahler polynomial
Meixner polynomial
Meixner-Pollaczek polynomial
Mott polynomial
Narumi polynomials
Neumann polynomial
Padovan polynomials
Peters polynomials
Pidduck polynomial
Pincherle polynomials
Sister Celine's polynomials
Spread polynomial
Touchard polynomial
Rook polynomial
Stirling polynomial


Basic Fourier series
Euler-Jackson $q$-difference operator
$q$-Bernoulli numbers
Symmetric $q$-numbers

$q$-special functions

Basic hypergeometric $\phi$
Basic hypergeometric series $\psi$
Continuous $q$-Hermite polynomials
Elliptic gamma function
Exton $q$-exponential
Generalized $q$-Bessel
Hahn-Exton $q$-Bessel
Gosper $q$-sine
Jackson $q$-Bessel (1)
Jackson $q$-Bessel (2)
Jackson $q$-Bessel (3)
LLT polynomials
$q$-Bessel function
$q$-Beta function
$q$-Binomial coefficient
$q$-Dirichlet series
$q$-exponential $e_q$
$q$-exponential $e_{\frac{1}{q}}$
$q$-exponential $E_q$
$q$-exponential $E_{\frac{1}{q}}$
$q$-Fibonacci polynomials
q-Gamma function
$q$-Gaussian distribution
$q$-Hermite polynomial
$q$-Hurwitz zeta
$q$-Pochhammer symbol
q-Pollaczek polynomial
$q$-Polygamma function
$q$-shifted factorial
$q$-Theta function
$q$-ultraspherical polynomial
Second $q$-shifted factorial
Tsallis $q$-exponential


Special functions in number theory

Number theory functions

Dirichlet $L$-function
$p$-adic $L$ function
Ramanujan's sum
Riemann prime counting

Special sequences

Sequences are merely functions whose domain is a subset of $\mathbb{Z}$.
Amicable numbers
Associated Stirling numbers of the second kind
Bell numbers
Bernoulli numbers ($B_0 = 1, B_1 = \pm \frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = −\frac{1}{30}, B_5 = 0, B_6 = \frac{1}{42}, B_7 = 0, B_8 = −\frac{1}{30}$)
Betrothed numbers
Carmichael numbers
Euler numbers
Erdős–Nicolas numbers
Farey sequence
Fermat numbers
Fibonacci numbers
Genocchi numbers
Harmonic number
Idoneal number
Jacobsthal numbers
Signed Lah numbers
Unsigned Lah numbers
Lucas numbers
Lychrel numbers
Pell numbers
Mersenne numbers
Nielsen-Ramanujan sequence
Perfect numbers
Powerful numbers
Pronic numbers
$q$-Bernoulli numbers
Reduced Stirling numbers of the second kind
Stieltjes constants
Stirling numbers of the first kind
Stirling numbers of the second kind
Sylvester's sequence
Thue-Morse sequence

Trigonometric functions

Arcsin (multivalued)
Arccos (multivalued)
Arctan (multivalued
Arccsc (multivalued)
Arcsec (multivalued)
Arccot (multivalued)

Trigonometric functions
Inverse trigonometric functions
Hyperbolic trigonometric functions
Inverse hyperbolic trigonometric functions
Depreciated trigonometric functions

Depreciated trigonometric functions

Inverse tangent integral

Zeta functions

A directory of zeta functions.

Airy zeta function
Arithmetic zeta function
Arakawa-Kaneko zeta function
Arithmetic zeta function
Artin-Mazur zeta function
Barnes zeta function
Bessel zeta function
Beurling zeta function
Cotangent zeta function
Dedekind zeta function
Epstein zeta function
Fibonacci 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
$q$-Hurwitz zeta
Reciprocal zeta function
Ruelle zeta function
Secant zeta function
Selberg zeta function
Shimizu zeta function
Shintani zeta function
Witten zeta function
Z function

Arithmetic Operations


Special constants


Name Notation Value
Apéry's constant $\zeta(3)$ $1.2020569031595942854 \ldots$
Artin constant $C_{\mathrm{Artin}}$ $0.3739558136\ldots$
Brun's constant $B_2$ $1.9021605831\ldots$
Cahen's constant $C$ $0.64341054629\ldots$
Catalan constant $G$ $0.915965594177219015054603514932384110774 \ldots$
Chaitin's constant
Champernowne constant $C_{10}$ $0.12345678910111213141516171819202122232425\ldots$
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 $4.669201609102990671853203820466201617258185577475768632745651 \ldots$
Freiman constant $F$ $4.5278295661\ldots$
Fransén–Robinson constant $2.807770242028519365221501186557772932308085920930198291220054 \ldots$
Gauss' constant $G$ $0.83462684167\ldots$
Gieseking constant $G$ $1.01494160640965\ldots$
Gelfond constant $e^{\pi}$ $23.14069263277926900572908636794854738026610624260021199344504 \ldots$
Gelfond–Schneider constant $2^{\sqrt{2}}$ $2.665144142690225188650297249873139848274211313714659492835979 \ldots$
Glaisher–Kinkelin constant $A$ $1.2824271291 \ldots$
Goh-Schmutz constant $C$ $1.1178641511899\ldots$
Golden ratio $\phi$ $1.618033988749894848204586834365638117720309179805762862135448 \ldots$
Gompertz constant $G$ $0.596347362323\ldots$
Graham's number
Hall-Montgomery constant $\delta_0$ $0.17150049\ldots$
Imaginary number $i$ $\sqrt{-1}$
Jenny's constants $J$ $867.5309\ldots$
Kaprekar's constant $6174$
Khinchin's constant $K$ $2.685452001065306445309714835481795693820382293994462953051152 \ldots$
Komornik–Loreti constant $q$ $1.787231650\ldots$
Legendre's constant $B$ $1$
Lefschetz number
Lévy's constant
Liouville constant
Meissel-Mertens constant $M$ $0.26149721284764278375...$
Mills' constant $M$ $1.306377883863080690468614492602605712916784585156713644368053 \ldots$
Nested radical constant $C$ $1.75793275\ldots$
Norton's constant $B$ $0.06535142\ldots$
Omega constant $\Omega$ $0.5671432904097838729\ldots$
Paper folding constant $P$ $0.85073618820186\ldots$
Pell constant $P$ $0.58057755820489\ldots$
Pi $\pi$ $3.141592653589793238462643383279502884197169399375105820974944 \ldots$
Porter's constant $C$ $1.4670780794\ldots$
q-pi $\pi_q$
Rabbit constant $R$ $0.7098034428612913146\ldots$
Ramanujan constant $R$ $262537412640768743.9999999999992500\ldots$
Reciprocal Fibonacci constant $\psi$ $3.35988566624317755\ldots$
Rutherford constant $K_R$ $0.8227\ldots$
Shallit Constant
Sierpiński constant $S$ $0.8228252496\ldots$
Silver ratio $\delta_s$ $2.4142135623730950488 \ldots$
Soldner's Constant $\mu$ $1.45136923488338105028396848589202744949\ldots$
Square root spiral constant $-2.1577829966\ldots$
Stieltjes constants $\gamma_n$
Ternary Champernowne constant $C_3$ $0.598958167538433\ldots$
Theodorus constant $1.8600250792\ldots$
Thue constant $0.8590997969\ldots$
Trott's constant $0.10841015\ldots$
Twin prime constant
White House switchboard constants $W$ $0.2024561414 \ldots$
Wyler constant $\alpha_W$
Zolotarev-Schur constant


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

Tools from calculus

Absolute maximum
Absolute minimum
Analytic continuation
Arithmetic functions
Binomial series
Difference equation of hypergeometric type
Doubly periodic function
Fundamental pair of periods
Integration by parts
Integral from a to a
Laplace transform
Lattice generated by doubly periodic periods
Lebesgue integral
Local maximum
Local minimum
Periodic function
Period parallelogram
Polar coordinates
Product rule for derivatives
Quotient rule
Ratio test
Riemann integral
Stirling formula
Rolle's theorem
Taylor series
Weierstrass factorization


Functions named after Pafnuty Chebyshev
Functions named after Peter Gustav Lejeune Dirichlet
Functions named after Leonard Euler
Functions named after Carl Gustav Jacob Jacobi
Functions named after Bernhard Riemann
Functions named after Karl Weierstrass

External Links

Higher transcendental functions (Bateman project)
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
T.H Koornwinder's list of resources
Victor Moll's proofs of Gradshteyn and Ryzhik integrals
LMFDB, the database of L-functions, modular forms, and related objects.
Phillipe Flajolet's "Analytic Combinatorics"
Online number theory lecture notes and teaching materials
The On-Line Encyclopedia of Integer Sequences
"According to Abramowitz and Stegun" or arccoth needn't be uncouth