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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.
Contents
- 1 How to get access to edit this Wiki
- 2 Special functions
- 3 Special constants
- 4 Operators
- 5 Numbers
- 6 Tools from calculus
- 7 Other
- 8 External Links
How to get access to edit this Wiki
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Special functions
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
Falling factorial
Floor
Heaviside step function
Hermite function
Incomplete beta function
Jacobi theta functions
K-function
Legendre chi
Legendre function
Lerch transcendent
Loggamma
Lower incomplete gamma
Mathieu function
Maass forms
Meijer G-function
Modified Struve function
Modular form
Mock modular forms
Mock theta functions
Multiple gamma function
$p$-adic $L$ function
Painlevé transcendents
Polyexponential
Ramanujan theta function
Riemann Siegel theta function
Riemann theta function
Rising factorial
Sievert integral
Signum
Struve function
Theta functions
Trigamma
Voight function
Weber function
Weierstrass elementary factors
Weierstrass function
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
Error functions
Dawson F
Inverse complementary error function
Bessel functions
Riccati-Bessel
Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$
Elliptic functions
Elliptic functions (general overview and definitions)
 Elliptic $E$ |
 Elliptic $K$ |
 Elliptic $\theta$ |
 Modular $\lambda$ |
 Weierstrass elliptic |
Incomplete Elliptic E
Incomplete Elliptic K
Jacobi elliptic functions
Weierstrass $\sigma$
Polynomials
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
Trigonometric functions
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
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
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
$q$-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
$q$-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
 $q$-Binomial |
 $q$-Gamma |
 $q$-factorial |
 Euler $\phi$ |
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
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
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 |
Operators
Alexander operator
Bernardi operator
Fourier transform
Laplace transform
Libera operator
Mellin transform
Radon transform
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
