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	Airy Bi	Anger function	Barnes $G$	Beta	Binomial
Dedekind eta	Devil's staircase	Digamma	Dilogarithm	Dirichlet eta	Dirichlet beta
Exponential	Factorial	Gamma	Hadamard gamma	Hyperfactorial	Hypergeometric ${}_pF_q$
Identity function	Upper incomplete gamma	Klein invariant $J$	Logarithm	Polygamma	Polylogarithm
Partition	Lambert $W$	Minkowski $?$	Planck radiation	Pochhammer	Ramanujan $\tau$
Reciprocal gamma	Riemann-Siegel $Z$	Riemann $\xi$	Scorer $\mathrm{Gi}$	Scorer $\mathrm{Hi}$	Thomae

Arithmetic functions
Automorphic forms and functions
Bessel-Clifford function
Böhmer integral
Buchstab function
Carlitz-Goss Gamma function
Ceiling
Clausen functions
Dawson function
Debye function
Dickman–de Bruijn function
Dirac delta
Dirichlet function
Distance to integers
Eisenstein series
Faddeeva function
Floor
Heaviside step function
Hermite function
Jacobi theta functions
K-function
Legendre chi
Legendre function
Lerch transcendent
Mathieu function
Maass forms
Meijer G-function
Modified Struve function
Modular form
Mock modular forms
Mock theta functions
$p$-adic $L$ function
Painlevé transcendents
Polyexponential
Ramanujan theta function
Riemann Siegel theta function
Riemann theta function
Sievert integral
Signum
Struve function
Theta functions
Voight function
Weber function
Weierstrass elementary factors
Weierstrass function

Bessel functions and friends

Bessel functions

Modified Bessel $K_{\nu}$

Spherical Bessel $j_{\nu}$

Spherical Bessel $y_{\nu}$

Kelvin functions

Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$

Continuous nowhere-differentiable functions

Besicovitch functions
Bolzano function
Cellérier function
Darboux function
Faber function F1
Faber function F2
Katsuura function
Keisswetter function
Koch curve
Knopp function
Lynch function
McCarthy function
Orlicz functions
Peano function
Petr function
Riemann function
Schoenberg function
Schwarz function
Sierpiński curve
Takagi function
van der Waerden function
Weierstrass function
Wen function

Elliptic functions

Elliptic functions (general overview and definitions)

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

Jacobi Elliptic Functions

Error function and friends

Dawson F
Inverse complementary error function

Error functions

Error function

Complementary $\mathrm{erf}$

Erfi

Inverse $\mathrm{erf}$

Figurate numbers

Tetrahedral numbers
Pyramidal numbers
Cubic numbers
Octahedral numbers

Polygonal numbers

Decagonal numbers
Dodecagonal numbers
Enneadecagonal numbers
Hendecagonal numbers
Heptadecagonal numbers
Heptagonal numbers
Hexadecagonal numbers
Hexagonal numbers
Icosagonal numbers
Icosidigonal numbers
Icositetragonal numbers
Icositrigonal numbers
Icosihenagonal numbers
Myriagonal numbers
Nonogonal numbers
Octadecagonal numbers
Octagonal numbers
Pentadecagonal numbers
Pentagonal numbers
Rectangular numbers
Square numbers
Thue-Morse sequence
Tridecagonal numbers
Triangular numbers
Tetradecagonal numbers

Operators

Alexander operator
Bernardi operator
Fourier transform
Laplace transform
Libera operator
Mellin transform
Radon transform

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

Angelescu polynomials
Bell polynomial
Boole polynomials
Charlier polynomial
Cyclotomic polynomials
Denisyuk polynomials
Hahn polynomial
Humbert polynomials
$q$-Hermite polynomial
Krawtchouk polynomial
Lagrange 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

$q$-calculus

Basic Fourier series
$q$-analog
$q$-Bernoulli numbers
$q$-derivative
$q$-numbers

$q$-special functions

Basic hypergeometric series $\phi$
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
$q$-$\cos$
$q$-$\mathrm{Cos}$
$q$-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
$q$-Hypergeometric function
$q$-Pochhammer symbol
q-Pollaczek polynomial
$q$-Polygamma function
$q$-shifted factorial
$q$-$\sin$
$q$-$\mathrm{Sin}$
$q$-Theta function
$q$-ultraspherical polynomial
Tsallis $q$-exponential

Special functions in number theory

Chebyshev $\vartheta$	Chebyshev $\psi$	Euler totient	Greatest prime factor	Mangoldt $\Lambda$	Mertens
Möbius	Liouville $\lambda$	Prime counting	Riemann zeta	Sum of divisors

Chebyshev $\vartheta$
Dirichlet $L$-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
Fermat numbers
Fibonacci sequence
Genocchi numbers
Harmonic number
Lah numbers
Lucas numbers
Lychrel numbers
Mersenne numbers
Nielsen-Ramanujan sequence
Perfect 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

Trigonometric functions

Inverse tangent integral

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

$*$-c functions

Coshc

Jinc

Sinc

Sinhc

Tanc

Tanhc

$\ast$-integral functions

Cosine integral

Cosh integral

Exp integral $E_n$

Exp integral $\mathrm{Ei}$

Fresnel $C$

Fresnel $S$

Gudermannian

Inverse $\mathrm{gd}$

Sine integral

Log integral

Sinh 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
Reciprocal zeta function
Ruelle zeta function
Secant zeta function
Selberg zeta function
Shimizu zeta function
Shintani zeta function
Witten zeta function

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
Catalan constant	$G$	$0.915965594177219015054603514932384110774 \ldots$
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		$4.669201609102990671853203820466201617258185577475768632745651 \ldots$
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	$K$	$2.685452001065306445309714835481795693820382293994462953051152 \ldots$
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	$B$	$0.06535142\ldots$
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
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
Stieltjes constants
Ternary Champernowne constant	$C_3$	$0.598958167538433\ldots$
Theodorus constant		$1.8600250792\ldots$
Thue constant		$0.8590997969\ldots$
Twin prime constant
White House switchboard constants	$W$	$0.2024561414 \ldots$
Wyler constant	$\alpha_W$
Zolotarev-Schur constant

Numbers

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
Cell
Continuous
Derivative
Difference equation of hypergeometric type
Doubly periodic function
Fundamental pair of periods
Integration by parts
Laplace transform
Lattice generated by doubly periodic periods
Lebesgue integral
Local maximum
Local minimum
Periodic function
Period parallelogram
Product rule
Quotient rule
Ratio test
Riemann integral
Stirling formula
Rolle's theorem
Taylor series
Weierstrass factorization

Other

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
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 Combinatorics"
Gallery of mathematical functions
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
OpenMath Content Dictionaries: Special Functions

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Contents

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Special functions

Bessel functions and friends

Continuous nowhere-differentiable functions

Elliptic functions

Error function and friends

Figurate numbers

Polygonal numbers

Gamma function and friends

Operators

Polynomials

$q$-calculus

$q$-special functions

Special functions in number theory

Special sequences

Trigonometric functions

Zeta functions

Special constants

Numbers

Tools from calculus

Other

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