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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

Bessel functions
Airy functions
Hankel functions
Kelvin functions

## *-c functions

$*$-c functions

## Continuous nowhere-differentiable functions and friends

Continuous nowhere differentiable functions

## Elliptic functions

Jacobi Elliptic Functions

Error functions

## Figurate numbers

Polygonal numbers

## Hypergeometric series

Hypergeometric functions

## *-integral functions

$\ast$-integral functions

## Jacobi theta functions

Jacobi theta functions

## Logarithm function and friends

Logarithm and friends

## Probability distributions

### Discrete distributions

Orthogonal polynomials

## $q$-calculus

### $q$-special functions

$q$-calculus

## Special functions in number theory

Number theory functions

## 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

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

## Zeta functions

A directory of zeta functions.

# 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

# Numbers

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