Difference between revisions of "Main Page"
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<div class="grid"><center>[[File:dedekindetathumb.png|45px|link=Dedekind eta]]<br /> | <div class="grid"><center>[[File:dedekindetathumb.png|45px|link=Dedekind eta]]<br /> | ||
[[Dedekind eta]]</center></div> | [[Dedekind eta]]</center></div> | ||
− | |||
− | |||
<div class="grid"><center>[[File:dirichletetathumb.png|45px|link=Dirichlet eta]]<br /> | <div class="grid"><center>[[File:dirichletetathumb.png|45px|link=Dirichlet eta]]<br /> | ||
[[Dirichlet eta]]</center></div> | [[Dirichlet eta]]</center></div> | ||
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<div class="grid"><center>[[File:lambertwthumb.png|45px|link=Lambert W]]<br /> | <div class="grid"><center>[[File:lambertwthumb.png|45px|link=Lambert W]]<br /> | ||
[[Lambert W|Lambert $W$]]</center></div> | [[Lambert W|Lambert $W$]]</center></div> | ||
− | |||
− | |||
<div class="grid"><center>[[File:planckradiationthumb.png|45px|link=Planck radiation]]<br /> | <div class="grid"><center>[[File:planckradiationthumb.png|45px|link=Planck radiation]]<br /> | ||
[[Planck radiation]]</center></div> | [[Planck radiation]]</center></div> | ||
<div class="grid"><center>[[File:ramanujantauthumb.png|45px|link=Ramanujan tau]]<br /> | <div class="grid"><center>[[File:ramanujantauthumb.png|45px|link=Ramanujan tau]]<br /> | ||
[[Ramanujan tau|Ramanujan $\tau$]]</center></div> | [[Ramanujan tau|Ramanujan $\tau$]]</center></div> | ||
− | |||
− | |||
<div class="grid"><center>[[File:riemannxithumb.png|45px|link=Riemann xi]]<br /> | <div class="grid"><center>[[File:riemannxithumb.png|45px|link=Riemann xi]]<br /> | ||
[[Riemann xi|Riemann $\xi$]]</center></div> | [[Riemann xi|Riemann $\xi$]]</center></div> | ||
Line 41: | Line 35: | ||
</div> | </div> | ||
− | |||
− | |||
− | |||
− | |||
[[Carlitz-Goss Gamma function]]<br /> | [[Carlitz-Goss Gamma function]]<br /> | ||
[[Clausen cosine]]<br /> | [[Clausen cosine]]<br /> | ||
[[Clausen sine]]<br /> | [[Clausen sine]]<br /> | ||
− | |||
− | |||
− | |||
[[Debye function]]<br /> | [[Debye function]]<br /> | ||
[[Determinant]]<br /> | [[Determinant]]<br /> | ||
− | |||
[[Dirichlet function]]<br /> | [[Dirichlet function]]<br /> | ||
[[Discriminant]]<br /> | [[Discriminant]]<br /> | ||
[[Distance to integers]]<br /> | [[Distance to integers]]<br /> | ||
[[Eisenstein series]]<br /> | [[Eisenstein series]]<br /> | ||
− | |||
− | |||
[[Heaviside step function]]<br /> | [[Heaviside step function]]<br /> | ||
[[Hermite function]]<br /> | [[Hermite function]]<br /> | ||
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[[Legendre function]]<br /> | [[Legendre function]]<br /> | ||
[[Lerch transcendent]]<br /> | [[Lerch transcendent]]<br /> | ||
− | [[ | + | [[Log barrier]]<br /> |
+ | [[MacRobert E]]<br /> | ||
[[Maass forms]] <br /> | [[Maass forms]] <br /> | ||
[[Meijer G-function]]<br /> | [[Meijer G-function]]<br /> | ||
− | |||
[[Modular form]]<br /> | [[Modular form]]<br /> | ||
[[Mock modular forms]]<br /> | [[Mock modular forms]]<br /> | ||
[[Mock theta functions]]<br /> | [[Mock theta functions]]<br /> | ||
[[Nome]]<br /> | [[Nome]]<br /> | ||
− | |||
[[Painlevé transcendents]]<br /> | [[Painlevé transcendents]]<br /> | ||
[[Polyexponential]]<br /> | [[Polyexponential]]<br /> | ||
[[Ramanujan theta function]]<br /> | [[Ramanujan theta function]]<br /> | ||
+ | [[Riemann-Landau xi]]<br /> | ||
[[Riemann Siegel theta function]]<br /> | [[Riemann Siegel theta function]]<br /> | ||
[[Riemann theta function]]<br /> | [[Riemann theta function]]<br /> | ||
[[Sievert integral]]<br /> | [[Sievert integral]]<br /> | ||
[[Singular function]]<br /> | [[Singular function]]<br /> | ||
− | |||
[[Theta functions]]<br /> | [[Theta functions]]<br /> | ||
+ | [[Unit step function]]<br /> | ||
[[Voight function]]<br /> | [[Voight function]]<br /> | ||
[[Volterra function]]<br /> | [[Volterra function]]<br /> | ||
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{{:Kelvin functions footer}} | {{:Kelvin functions footer}} | ||
+ | |||
+ | |||
+ | [[Modified Struve function]]<br /> | ||
[[Riccati-Bessel S | Riccati-Bessel $S_n$]]<br /> | [[Riccati-Bessel S | Riccati-Bessel $S_n$]]<br /> | ||
[[Riccati-Bessel C | Riccati-Bessel $C_n$]]<br /> | [[Riccati-Bessel C | Riccati-Bessel $C_n$]]<br /> | ||
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[[Riccati-Bessel Zeta | Riccati-Bessel $\zeta_n$]] <br /> | [[Riccati-Bessel Zeta | Riccati-Bessel $\zeta_n$]] <br /> | ||
[[Weber function]]<br /> | [[Weber function]]<br /> | ||
+ | <div class="grid2"><center>[[Struve function|<randomimagebycategory categories="Struveglyph" width="45" />]]<br /> | ||
+ | [[Struve function]]</center></div> | ||
==*-c functions== | ==*-c functions== | ||
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==Continuous nowhere-differentiable functions and friends== | ==Continuous nowhere-differentiable functions and friends== | ||
<div class="grid"> | <div class="grid"> | ||
+ | <div class="grid"><center>[[File:devilstaircasethumb.png|45px|link=Devil's staircase]]<br /> | ||
+ | [[Devil's staircase]]</center></div> | ||
+ | <div class="grid"><center>[[File:minkowskiqmthumb.png|45px|link=Minkowski question mark]]<br /> | ||
+ | [[Minkowski question mark|Minkowski $?$]]</center></div> | ||
<div class="grid"><center>[[Riemann function|<randomimagebycategory categories="riemannglyph" width=45 />]]<br /> | <div class="grid"><center>[[Riemann function|<randomimagebycategory categories="riemannglyph" width=45 />]]<br /> | ||
[[Riemann function]]</center></div> | [[Riemann function]]</center></div> | ||
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[[Weierstrass nowhere differentiable function]]<br /> | [[Weierstrass nowhere differentiable function]]<br /> | ||
[[Wen function]]<br /> | [[Wen function]]<br /> | ||
+ | |||
+ | ==Defined by a differential equation== | ||
+ | [[Buchstab function]]<br /> | ||
+ | [[Dickman–de Bruijn function]]<br /> | ||
+ | [[Mathieu function]] <br /> | ||
==Elliptic functions== | ==Elliptic functions== | ||
Line 157: | Line 155: | ||
==Error function and friends== | ==Error function and friends== | ||
− | [[Dawson | + | <div class="grid2"><center>[[Dawson D+|<randomimagebycategory categories="dawsondplusglyph" width="45" />]]<br /> |
+ | [[Dawson D+]]</center></div> | ||
+ | [[Dawson D-]]<br /> | ||
+ | [[Faddeeva function]]<br /> | ||
[[Inverse complementary error function]]<br /> | [[Inverse complementary error function]]<br /> | ||
{{:Error functions footer}} | {{:Error functions footer}} | ||
Line 220: | Line 221: | ||
{{:Hypergeometric functions footer}} | {{:Hypergeometric functions footer}} | ||
+ | [[Hypergeometric 0F3]]<br /> | ||
[[Hypergeometric 1F1]]<br /> | [[Hypergeometric 1F1]]<br /> | ||
[[Hypergeometric 1F2]]<br /> | [[Hypergeometric 1F2]]<br /> | ||
[[Hypergeometric 2F0]]<br /> | [[Hypergeometric 2F0]]<br /> | ||
+ | [[Hypergeometric 2F3]]<br /> | ||
+ | [[Hypergeometric 3F2]]<br /> | ||
+ | [[Hypergeometric 4F1]]<br /> | ||
+ | [[Lauricella–Saran functions]]<br /> | ||
==*-integral functions== | ==*-integral functions== | ||
Line 231: | Line 237: | ||
==Jacobi theta functions== | ==Jacobi theta functions== | ||
− | + | {{:Jacobi theta footer}} | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
==Logarithm function and friends== | ==Logarithm function and friends== | ||
− | + | {{:Logarithm and friends footer}} | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
+ | [[Logarithm base a]]<br /> | ||
[[Logarithm (multivalued)]]<br /> | [[Logarithm (multivalued)]]<br /> | ||
[[Trilogarithm]]<br /> | [[Trilogarithm]]<br /> | ||
+ | |||
+ | ==Matrix functions== | ||
+ | [[Determinant]] <br /> | ||
+ | [[Matrix arcsin]]<br /> | ||
+ | [[Matrix exponential]]<br /> | ||
+ | [[Matrix cosh]]<br /> | ||
+ | [[Matrix sinh]]<br /> | ||
+ | [[Matrix logarithm]]<br /> | ||
+ | [[Matrix sine]]<br /> | ||
+ | [[Matrix cosine]]<br /> | ||
+ | [[Trace]]<br /> | ||
==Operators== | ==Operators== | ||
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[[Mellin transform]]<br /> | [[Mellin transform]]<br /> | ||
[[Radon transform]]<br /> | [[Radon transform]]<br /> | ||
+ | [[Two-dimensional Laplace transform]]<br /> | ||
==Polynomials== | ==Polynomials== | ||
Line 271: | Line 272: | ||
[[Abel p|Abel $p$]]</center></div> | [[Abel p|Abel $p$]]</center></div> | ||
</div> | </div> | ||
+ | [[Bernstein B]]<br /> | ||
+ | |||
+ | ==Probability distributions== | ||
+ | ===Continuous distributions=== | ||
+ | [[Arcsin distribution]]<br /> | ||
+ | [[Beta distribution]]<br /> | ||
+ | [[Cauchy distribution]]<br /> | ||
+ | [[Chi-squared distribution]]<br /> | ||
+ | [[Exponential distribution]]<br /> | ||
+ | [[F distribution]]<br /> | ||
+ | [[Gamma distribution]]<br /> | ||
+ | [[Laplace distribution]]<br /> | ||
+ | [[Log-normal distribution]]<br /> | ||
+ | [[Normal distribution]]<br /> | ||
+ | [[Pareto distribution]]<br /> | ||
+ | [[Student's t distribution]]<br /> | ||
+ | [[Uniform distribution]]<br /> | ||
+ | [[Weibull distribution]]<br /> | ||
+ | |||
+ | ===Discrete distributions=== | ||
+ | [[Bernoulli distribution]]<br /> | ||
+ | [[Binomial distribution]]<br /> | ||
+ | [[Discrete uniform distribution]]<br /> | ||
+ | [[Geometric distribution]]<br /> | ||
+ | [[Hypergeometric distribution]]<br /> | ||
+ | [[Negative binomial distribution]]<br /> | ||
+ | [[Poisson distribution]]<br /> | ||
+ | |||
{{:Orthogonal polynomials footer}} | {{:Orthogonal polynomials footer}} | ||
− | + | [[Adomian polynomials]]<br /> | |
[[Angelescu polynomials]]<br /> | [[Angelescu polynomials]]<br /> | ||
[[Bell polynomial]]<br /> | [[Bell polynomial]]<br /> | ||
Line 315: | Line 344: | ||
===$q$-special functions=== | ===$q$-special functions=== | ||
− | [[Basic hypergeometric | + | [[Basic hypergeometric phi|Basic hypergeometric $\phi$]]<br /> |
[[Basic hypergeometric series psi|Basic hypergeometric series $\psi$]]<br /> | [[Basic hypergeometric series psi|Basic hypergeometric series $\psi$]]<br /> | ||
[[Continuous q-Hermite polynomial | Continuous $q$-Hermite polynomials]]<br /> | [[Continuous q-Hermite polynomial | Continuous $q$-Hermite polynomials]]<br /> | ||
Line 341: | Line 370: | ||
[[q-Gaussian distribution | $q$-Gaussian distribution]]<br /> | [[q-Gaussian distribution | $q$-Gaussian distribution]]<br /> | ||
[[q-Hermite polynomial | $q$-Hermite polynomial]]<br /> | [[q-Hermite polynomial | $q$-Hermite polynomial]]<br /> | ||
− | |||
[[q-Pochhammer | $q$-Pochhammer symbol]]<br /> | [[q-Pochhammer | $q$-Pochhammer symbol]]<br /> | ||
[[q-Pollaczek polynomial]]<br /> | [[q-Pollaczek polynomial]]<br /> | ||
Line 359: | Line 387: | ||
[[Dirichlet L-function | Dirichlet $L$-function]]<br /> | [[Dirichlet L-function | Dirichlet $L$-function]]<br /> | ||
+ | [[p-adic L function|$p$-adic $L$ function]]<br /> | ||
[[Ramanujan's sum]]<br /> | [[Ramanujan's sum]]<br /> | ||
[[Riemann prime counting]]<br /> | [[Riemann prime counting]]<br /> | ||
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[[Farey sequence]]<br /> | [[Farey sequence]]<br /> | ||
[[Fermat numbers]]<br /> | [[Fermat numbers]]<br /> | ||
− | [[Fibonacci | + | [[Fibonacci numbers]]<br /> |
[[Genocchi numbers]]<br /> | [[Genocchi numbers]]<br /> | ||
[[Harmonic number]]<br /> | [[Harmonic number]]<br /> | ||
Line 380: | Line 409: | ||
[[Lucas numbers]]<br /> | [[Lucas numbers]]<br /> | ||
[[Lychrel numbers]]<br /> | [[Lychrel numbers]]<br /> | ||
+ | [[Pell numbers]]<br /> | ||
[[Mersenne numbers]]<br /> | [[Mersenne numbers]]<br /> | ||
[[Nielsen-Ramanujan sequence]]<br /> | [[Nielsen-Ramanujan sequence]]<br /> | ||
Line 390: | Line 420: | ||
[[Stirling numbers of the first kind]]<br /> | [[Stirling numbers of the first kind]]<br /> | ||
[[Stirling numbers of the second kind]]<br /> | [[Stirling numbers of the second kind]]<br /> | ||
+ | [[Sylvester's sequence]]<br /> | ||
[[Thue-Morse sequence]]<br /> | [[Thue-Morse sequence]]<br /> | ||
Line 418: | Line 449: | ||
[[Haversine]]<br /> | [[Haversine]]<br /> | ||
[[Inverse tangent integral]]<br /> | [[Inverse tangent integral]]<br /> | ||
+ | [[Vercosine]]<br /> | ||
==Zeta functions== | ==Zeta functions== | ||
Line 489: | Line 521: | ||
| [[Brun's constant]] || $B_2$ || $1.9021605831\ldots$ | | [[Brun's constant]] || $B_2$ || $1.9021605831\ldots$ | ||
|- | |- | ||
− | | [[Cahen's constant]] || | + | | [[Cahen's constant]] || $C$ || $0.64341054629\ldots$ |
|- | |- | ||
| [[Catalan constant]] || $G$ || $0.915965594177219015054603514932384110774 \ldots$ | | [[Catalan constant]] || $G$ || $0.915965594177219015054603514932384110774 \ldots$ | ||
Line 540: | Line 572: | ||
|- | |- | ||
| [[Khinchin's constant]] || $K$ || $2.685452001065306445309714835481795693820382293994462953051152 \ldots$ | | [[Khinchin's constant]] || $K$ || $2.685452001065306445309714835481795693820382293994462953051152 \ldots$ | ||
+ | |- | ||
+ | | [[Komornik–Loreti constant]] || $q$ || $1.787231650\ldots$ | ||
|- | |- | ||
| [[Legendre's constant]] || $B$ || $1$ | | [[Legendre's constant]] || $B$ || $1$ | ||
Line 573: | Line 607: | ||
| [[Ramanujan constant]] || $R$ || $262537412640768743.9999999999992500\ldots$ | | [[Ramanujan constant]] || $R$ || $262537412640768743.9999999999992500\ldots$ | ||
|- | |- | ||
− | | [[Reciprocal Fibonacci constant]] || $\psi$ || | + | | [[Reciprocal Fibonacci constant]] || $\psi$ || $3.35988566624317755\ldots$ |
|- | |- | ||
| [[Rutherford constant]] || $K_R$ || $0.8227\ldots$ | | [[Rutherford constant]] || $K_R$ || $0.8227\ldots$ | ||
Line 621: | Line 655: | ||
[[Absolute minimum]]<br /> | [[Absolute minimum]]<br /> | ||
[[Analytic continuation]]<br /> | [[Analytic continuation]]<br /> | ||
+ | [[Arithmetic functions]]<br /> | ||
[[Binomial series]]<br /> | [[Binomial series]]<br /> | ||
[[Cell]]<br /> | [[Cell]]<br /> | ||
Line 627: | Line 662: | ||
[[Difference equation of hypergeometric type]]<br /> | [[Difference equation of hypergeometric type]]<br /> | ||
[[Doubly periodic function]]<br /> | [[Doubly periodic function]]<br /> | ||
+ | [[Entire]]<br /> | ||
[[Fundamental pair of periods]]<br /> | [[Fundamental pair of periods]]<br /> | ||
[[Integration by parts]]<br /> | [[Integration by parts]]<br /> | ||
Line 661: | Line 697: | ||
[http://www.johndcook.com/special_function_diagram.html Relations between special functions by John D. Cook]<br /> | [http://www.johndcook.com/special_function_diagram.html Relations between special functions by John D. Cook]<br /> | ||
[http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]<br /> | [http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]<br /> | ||
− | |||
[https://staff.fnwi.uva.nl/t.h.koornwinder/specfun/ T.H Koornwinder's list of resources]<br /> | [https://staff.fnwi.uva.nl/t.h.koornwinder/specfun/ T.H Koornwinder's list of resources]<br /> | ||
[http://129.81.170.14/~vhm/Table.html Victor Moll's proofs of Gradshteyn and Ryzhik integrals]<br /> | [http://129.81.170.14/~vhm/Table.html Victor Moll's proofs of Gradshteyn and Ryzhik integrals]<br /> | ||
− | |||
[http://www.lmfdb.org/ LMFDB, the database of L-functions, modular forms, and related objects.]<br /> | [http://www.lmfdb.org/ LMFDB, the database of L-functions, modular forms, and related objects.]<br /> | ||
[http://algo.inria.fr/flajolet/Publications/books.html Phillipe Flajolet's "Analytic Combinatorics"]<br /> | [http://algo.inria.fr/flajolet/Publications/books.html Phillipe Flajolet's "Analytic Combinatorics"]<br /> |
Revision as of 12:22, 7 November 2017
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 Special functions
- 1.1 Bessel functions and friends
- 1.2 *-c functions
- 1.3 Continuous nowhere-differentiable functions and friends
- 1.4 Defined by a differential equation
- 1.5 Elliptic functions
- 1.6 Error function and friends
- 1.7 Figurate numbers
- 1.8 Gamma function and friends
- 1.9 Hypergeometric series
- 1.10 *-integral functions
- 1.11 Jacobi theta functions
- 1.12 Logarithm function and friends
- 1.13 Matrix functions
- 1.14 Operators
- 1.15 Polynomials
- 1.16 Probability distributions
- 1.17 $q$-calculus
- 1.18 Special functions in number theory
- 1.19 Special sequences
- 1.20 Trigonometric functions
- 1.21 Zeta functions
- 1.22 Arithmetic Operations
- 2 Special constants
- 3 Numbers
- 4 Tools from calculus
- 5 Other
- 6 External Links
Special functions
Carlitz-Goss Gamma function
Clausen cosine
Clausen sine
Debye function
Determinant
Dirichlet function
Discriminant
Distance to integers
Eisenstein series
Heaviside step function
Hermite function
K-function
Legendre chi
Legendre function
Lerch transcendent
Log barrier
MacRobert E
Maass forms
Meijer G-function
Modular form
Mock modular forms
Mock theta functions
Nome
Painlevé transcendents
Polyexponential
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
Weierstrass function
Bessel functions and friends
Modified Struve function
Riccati-Bessel $S_n$
Riccati-Bessel $C_n$
Riccati-Bessel $\xi_n$
Riccati-Bessel $\zeta_n$
Weber function
*-c functions
Continuous nowhere-differentiable functions and friends
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)
Incomplete Elliptic E
Incomplete Elliptic K
Jacobi elliptic functions
Weierstrass $\sigma$
Error function and friends
Dawson D-
Faddeeva function
Inverse complementary error function
Figurate numbers
Tetrahedral numbers
Pyramidal numbers
Cubic numbers
Octahedral 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
Loggamma
Lower incomplete gamma
Multiple gamma function
Primorial
Rising factorial
Trigamma
Hypergeometric series
Hypergeometric 0F3
Hypergeometric 1F1
Hypergeometric 1F2
Hypergeometric 2F0
Hypergeometric 2F3
Hypergeometric 3F2
Hypergeometric 4F1
Lauricella–Saran functions
*-integral functions
Jacobi theta functions
Logarithm function and friends
Logarithm base a
Logarithm (multivalued)
Trilogarithm
Matrix functions
Determinant
Matrix arcsin
Matrix exponential
Matrix cosh
Matrix sinh
Matrix logarithm
Matrix sine
Matrix cosine
Trace
Operators
Alexander operator
Bernardi operator
Fourier transform
Laplace transform
Libera operator
Mellin transform
Radon transform
Two-dimensional Laplace transform
Polynomials
Probability distributions
Continuous distributions
Arcsin distribution
Beta distribution
Cauchy distribution
Chi-squared distribution
Exponential distribution
F distribution
Gamma distribution
Laplace distribution
Log-normal distribution
Normal distribution
Pareto distribution
Student's t distribution
Uniform distribution
Weibull distribution
Discrete distributions
Bernoulli distribution
Binomial distribution
Discrete uniform distribution
Geometric distribution
Hypergeometric distribution
Negative binomial distribution
Poisson distribution
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
$q$-calculus
Basic Fourier series
Euler-Jackson $q$-difference operator
$q$-analog
$q$-Bernoulli numbers
$q$-derivative
$q$-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
$q$-$\cos$
$q$-$\mathrm{Cos}$
$q$-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
$q$-Pochhammer symbol
q-Pollaczek polynomial
$q$-Polygamma function
$q$-shifted factorial
$q$-$\sin$
$q$-$\mathrm{Sin}$
$q$-Theta function
$q$-ultraspherical polynomial
Second $q$-shifted factorial
Tsallis $q$-exponential
Special functions in number theory
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
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)
Depreciated trigonometric functions
Arccosvercosine
Arccoversine
Archavercosine
Archaversine
Arcvercosine
Arcversine
Hacovercosine
Hacoversine
Havercosine
Haversine
Inverse tangent integral
Vercosine
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
Z function
Arithmetic Operations
Addition
Division
Exponentiation
Hyperoperation
Multiplication
Subtraction
Tetration
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 | $0.8590997969\ldots$ | |
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
Arithmetic functions
Binomial series
Cell
Continuous
Derivative
Difference equation of hypergeometric type
Doubly periodic function
Entire
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
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
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