Latest revision as of 13:31, 8 November 2024

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 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.
-=How to get access to edit this Wiki=
+<b><u>Registration</u></b>
-We have had a big problem with auto-registering bots, so please send an e-mail to tomcuchta....at....gmail....dot....com with subject "Special function wiki registration" and I will enable registration for you.
+Due to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki.
 =Special functions=
-<table border="0">
+<div class="grid">
-<tr>
+<div class="grid"><center>[[Ceiling|<randomimagebycategory categories="ceilingglyph" width="45" />]] <br />
-<td><center>[[File:Airyaithumb.png|45px|link=Airy Ai]]<br />
+[[Ceiling]]</center> </div>
-[[Airy Ai]]</center></td>
+<div class="grid"><center>[[File:dedekindetathumb.png|45px|link=Dedekind eta]]<br />
-<td><center>[[File:Airybithumb.png|45px|link=Airy Bi]]<br />
+[[Dedekind eta]]</center></div>
-[[Airy Bi]]</center></td>
+<div class="grid"><center>[[File:dirichletetathumb.png|45px|link=Dirichlet eta]]<br />
-<td><center>[[File:barnesgthumb.png|45px|link=Barnes G]]<br />
+[[Dirichlet eta]]</center></div>
-[[Barnes G|Barnes $G$]]</center></td>
+<div class="grid"><center>[[File:dirichletbetathumb.png|45px|link=Dirichlet beta]]<br />
-<td><center>[[File:betathumb.png|45px|link=Beta]]<br />
+[[Dirichlet beta]]</center></div>
-[[Beta]]</center></td>
+<div class="grid"><center>[[Exponential|<randomimagebycategory categories="exponentialglyph" width="45" />]]<br />
-<td><center>[[File:binomialthumb.png|45px|link=Binomial coefficient]]<br />
+[[Exponential]]</center></div>
-[[Binomial coefficient|Binomial]]</center></td>
+<div class="grid"><center>[[Floor|<randomimagebycategory categories="floorglyph" width="45" />]]<br />
-<td><center>[[File:Blancmangethumb.png|45px|link=Blancmange function]]<br />
+[[Floor]]</center></div>
-[[Blancmange function]]</center></td>
+<div class="grid"><center>[[Identity function|<randomimagebycategory categories="identityglyph" width="45" />]]<br />
-</tr>
+[[Identity function|Identity]]</center></div>
-<tr>
+<div class="grid"><center>[[File:kleininvariantjthumb.png|45px|link=Klein invariant J]]<br />
-<td><center>[[File:dedekindetathumb.png|45px|link=Dedekind eta]]<br />
+[[Klein invariant J|Klein invariant $J$]]</center></div>
-[[Dedekind eta]]</center></td>
+<div class="grid"><center>[[File:partitionthumb.png|45px|link=Partition]]<br />
-<td><center>[[File:devilstaircasethumb.png|45px|link=Devil's staircase]]<br />
+[[Partition]]</center></div>
-[[Devil's staircase]]</center></td>
+<div class="grid"><center>[[File:lambertwthumb.png|45px|link=Lambert W]]<br />
-<td><center>[[File:digammathumb.png|45px|link=Digamma]]<br />
+[[Lambert W|Lambert $W$]]</center></div>
-[[Digamma]]</center></td>
+<div class="grid"><center>[[File:planckradiationthumb.png|45px|link=Planck radiation]]<br />
-<td><center>[[File:dirichletetathumb.png|45px|link=Dirichlet eta]]<br />
+[[Planck radiation]]</center></div>
-[[Dirichlet eta]]</center></td>
+<div class="grid"><center>[[File:ramanujantauthumb.png|45px|link=Ramanujan tau]]<br />
-<td><center>[[File:dirichletbetathumb.png|45px|link=Dirichlet beta]]<br />
+[[Ramanujan tau|Ramanujan $\tau$]]</center></div>
-[[Dirichlet beta]]</center></td>
+<div class="grid"><center>[[File:riemannxithumb.png|45px|link=Riemann xi]]<br />
-<td><center>[[File:erfthumb.png|45px|link=Error function]]<br />
+[[Riemann xi|Riemann $\xi$]]</center></div>
-[[Error function]]</center></td>
+<div class="grid"><center>[[Signum|<randomimagebycategory categories="Signumglyph" width="45" />]]<br />
-</tr>
+[[Signum]]</center></div>
-<tr>
+<div class="grid"><center>[[File:thomaethumb.png|45px|link=Thomae]]<br />
-<td><center>[[File:expthumb.png|45px|link=Exponential]]<br />
+[[Thomae]]</center></div>
-[[Exponential]]</center></td>
+</div>
-<td><center>[[File:eithumb.png|45px|link=Exponential integral]]<br />
-[[Exponential integral]]</center></td>
-<td><center>[[File:factorialthumb.png|45px|link=Factorial]]<br />
-[[Factorial]]</center></td>
-<td><center>[[File:gammathumb.png|45px|link=Gamma]]<br />
-[[Gamma]]</center></td>
-<td><center>[[File:hadamardgammathumb.png|45px|link=Hadamard gamma]]<br />
-[[Hadamard gamma]]</center></td>
-<td><center>[[File:hyperfactorialthumb.png|45px|link=Hyperfactorial]]<br />
-[[Hyperfactorial]]</center></td>
-</tr>
-<tr>
-<td><center>[[File:hypergeometricthumb.png|45px|link=Hypergeometric pFq]]<br />
-[[Hypergeometric pFq|Hypergeometric ${}_pF_q$]]</center></td>
-<td><center>[[File:Incompletegammathumb.png|45px|link=Upper incomplete gamma]]<br />
-[[Upper incomplete gamma]]</center></td>
-<td><center>[[File:kleininvariantjthumb.png|45px|link=Klein invariant J]]<br />
-[[Klein invariant J|Klein invariant $J$]]</center></td>
-<td><center>[[File:lithumb.png|45px|link=Logarithmic integral]]<br />
-[[Logarithmic integral|Log integral]]</center></td>
-<td><center>[[File:logthumb.png|45px|link=Logarithm]]<br />
-[[Logarithm]]</center></td>
-<td><center>[[File:thomaethumb.png|45px|link=Thomae]]<br />
-[[Thomae]]</center></td>
-</tr>
-<tr>
-<td><center>[[File:digammathumb.png|45px|link=Polygamma]]<br />
-[[Polygamma]]</center></td>
-<td><center>[[File:polylog1thumb.png|45px|link=Polylogarithm]]<br />
-[[Polylogarithm]]</center></td>
-<td><center>[[File:partitionthumb.png|45px|link=Partition]]<br />
-[[Partition]]</center></td>
-<td><center>[[File:reciprocalgammathumb.png|45px|link=Reciprocal gamma]]<br />
-[[Reciprocal gamma]]</center></td>
-<td><center>[[File:lambertwthumb.png|45px|link=Lambert W]]<br />
-[[Lambert W|Lambert $W$]]</center></td>
-<td><center>[[File:minkowskiqmthumb.png|45px|link=Minkowski question mark]]<br />
-[[Minkowski question mark|Minkowski $?$]]</center></td>
-</tr>
-<tr>
-<td><center>[[File:planckradiationthumb.png|45px|link=Planck radiation]]<br />
-[[Planck radiation]]</center></td>
-<td><center>[[File:pochhammerthumb.png|45px|link=Pochhammer]]<br />
-[[Pochhammer]]</center></td>
-<td><center>[[File:ramanujantauthumb.png|45px|link=Ramanujan tau]]<br />
-[[Ramanujan tau|Ramanujan $\tau$]]</center></td>
-<td><center>[[File:riemannsiegelzthumb.png|45px|link=Riemann-Siegel Z]]<br />
-[[Riemann-Siegel Z|Riemann-Siegel $Z$]]</center></td>
-<td><center>[[File:riemannxithumb.png|45px|link=Riemann xi]]<br />
-[[Riemann xi|Riemann $\xi$]]</center></td>
-</tr>
-</table>
-[[Automorphic forms and functions]]<br />
+[[Bickley-Naylor]]<br />
 [[Carlitz-Goss Gamma function]]<br />
-[[Clausen functions]]<br />
+[[Clausen cosine]]<br />
-[[Dawson function]]<br />
+[[Clausen sine]]<br />
+[[Debye function]]<br />
+[[Dirichlet function]]<br />
+[[Discriminant]]<br />
+[[Distance to integers]]<br />
 [[Eisenstein series]]<br />
 [[Heaviside step function]]<br />
 [[Hermite function]]<br />
-[[Jacobi theta functions]]<br />
 [[K-function]]<br />
 [[Legendre chi]]<br />
 [[Legendre function]]<br />
 [[Lerch transcendent]]<br />
-[[Mathieu function]] <br />
+[[Log barrier]]<br />
+[[MacRobert E]]<br />
 [[Maass forms]] <br />
 [[Meijer G-function]]<br />
+[[Mittag-Leffler]]<br />
 [[Modular form]]<br />
 [[Mock modular forms]]<br />
 [[Mock theta functions]]<br />
-[[p-adic L function|$p$-adic $L$ function]]<br />
+[[Nome]]<br />
 [[Painlevé transcendents]]<br />
+[[Polyexponential]]<br />
+[[Ramanujan theta function]]<br />
+[[Riemann-Landau xi]]<br />
+[[Riemann Siegel theta function]]<br />
 [[Riemann theta function]]<br />
 [[Sievert integral]]<br />
-[[Signum]]<br />
+[[Singular function]]<br />
-[[Struve functions]]<br />
 [[Theta functions]]<br />
+[[Unit step function]]<br />
+[[Voight function]]<br />
+[[Volterra function]]<br />
+[[Weierstrass elementary factors]]<br />
+==Bessel functions and friends==
+{{:Bessel functions footer}}
+{{:Airy functions footer}}
+{{:Hankel functions footer}}
+{{:Kelvin functions footer}}
+[[Modified Struve function]]<br />
+[[Riccati-Bessel S | Riccati-Bessel $S_n$]]<br />
+[[Riccati-Bessel C | Riccati-Bessel $C_n$]]<br />
+[[Riccati-Bessel Xi | Riccati-Bessel $\xi_n$]] <br />
+[[Riccati-Bessel Zeta | Riccati-Bessel $\zeta_n$]] <br />
+[[Weber function]]<br />
+<div class="grid2"><center>[[Struve function|<randomimagebycategory categories="Struveglyph" width="45" />]]<br />
+[[Struve function]]</center></div>
+==*-c functions==
+[[Normalized sinc]]<br />
+{{:*-c functions footer}}
+==Continuous nowhere-differentiable functions and friends==
+<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 />
+[[Riemann function]]</center></div>
+</div>
+{{:Continuous nowhere differentiable functions footer}}
-==Bessel functions==
+[[Besicovitch functions]]<br />
-<table><tr>
+[[Bolzano function]]<br />
-<td><center>[[File:besselj0thumb.png|45px|link=Bessel]]<br />
+[[Cellérier function]]<br />
-[[Bessel J sub nu|Bessel $J_{\nu}$]]</center></td>
+[[Katsuura function]]<br />
-</tr></table>
+[[Keisswetter function]]<br />
+[[Koch curve]] <br />
+[[Knopp function]]<br />
+[[Lynch function]]<br />
+[[McCarthy function]]<br />
+[[Orlicz functions]]<br />
+[[Peano function]]<br />
+[[Petr function]]<br />
+[[Schoenberg function]]<br />
+[[Schwarz function]]<br />
+[[Sierpiński curve]]<br />
+[[Weierstrass nowhere differentiable function]]<br />
+[[Wen function]]<br />
-[[Bessel Y sub nu|Bessel $Y_{\nu}$]]<br />
+==Defined by a differential equation==
-[[Hankel H sub nu (1) | Hankel $H_{\nu}^{(1)}$]]<br />
+[[Buchstab function]]<br />
-[[Hankel H sub nu (2) | Hankel $H_{\nu}^{(2)}$]]<br />
+[[Dickman–de Bruijn function]]<br />
-[[Modified Bessel I sub nu | Modified Bessel $I_{\nu}$]]<br />
+[[Mathieu function]] <br />
-[[Modified Bessel K sub nu | Modified Bessel $K_{\nu}$]]<br />
-[[Riccati-Bessel S sub n | Riccati-Bessel $S_n$]]<br />
-[[Riccati-Bessel C sub n | Riccati-Bessel $C_n$]]<br />
-[[Riccati-Bessel Xi sub n | Riccati-Bessel $\xi_n$]] <br />
-[[Riccati-Bessel Zeta sub n | Riccati-Bessel $\zeta_n$]] <br />
-[[Spherical Bessel j sub nu | Spherical Bessel $j_{\nu}$]]<br />
-[[Spherical Bessel y sub nu | Spherical Bessel $y_{\nu}$]]<br />
-[[Spherical Hankel h sub nu (1) | Spherical Hankel $h_{\nu}^{(1)}$]]<br />
-[[Spherical Hankel h sub nu (2) | Spherical Hankel $h_{\nu}^{(2)}$]]<br />
 ==Elliptic functions==
-<table border="0">
+[[Elliptic function|Elliptic functions (general overview and definitions)]]
-<tr>
-<td><center>[[File:ellipticethumb.png|45px|link=Elliptic E]]<br />
+<div class="grid">
-[[Elliptic E|Elliptic $E$]]</center></td>
+<div class="grid"><center>[[File:ellipticethumb.png|45px|link=Elliptic E]]<br />
-<td><center>[[File:elliptickthumb.png|45px|link=Elliptic K]]<br />
+[[Elliptic E|Elliptic $E$]]</center></div>
-[[Elliptic K|Elliptic $K$]]</center></td>
+<div class="grid"><center>[[File:elliptickthumb.png|45px|link=Elliptic K]]<br />
-<td><center>[[File:ellipticthetathumb.png|45px|link=Elliptic theta]]<br />
+[[Elliptic K|Elliptic $K$]]</center></div>
-[[Elliptic theta|Elliptic $\theta$]]</center></td>
+<div class="grid"><center>[[File:ellipticthetathumb.png|45px|link=Elliptic theta]]<br />
-<td><center>[[File:jacobicnthumb.png|45px|link=Jacobi cn]]<br />
+[[Elliptic theta|Elliptic $\theta$]]</center></div>
-[[Jacobi cn|Jacobi $\mathrm{cn}$]]</center></td>
+<div class="grid"><center>[[File:modularlambdathumb.png|45px|link=Modular lambda]]<br />
-<td><center>[[File:jacobicdthumb.png|45px|link=Jacobi cd]]<br />
+[[Modular lambda|Modular $\lambda$]]</center></div>
-[[Jacobi cd|Jacobi $\mathrm{cd}$]]</center></td>
+<div class="grid"><center>[[File:weierstrasspthumb.png|45px|link=Weierstrass elliptic]]<br />
-<td><center>[[File:jacobicsthumb.png|45px|link=Jacobi cs]]<br />
+[[Weierstrass elliptic|Weierstrass elliptic]]</center></div>
-[[Jacobi cs|Jacobi $\mathrm{cs}$]]</center></td>
+</div>
-<td><center>[[File:jacobidcthumb.png|45px|link=Jacobi dc]]<br />
-[[Jacobi dc|Jacobi $\mathrm{dc}$]]</center></td>
+{{:Jacobi elliptic functions footer}}
-</tr>
-<tr>
+[[Incomplete Elliptic E]]<br />
-<td><center>[[File:jacobidnthumb.png|45px|link=Jacobi dn]]<br />
+[[Incomplete Elliptic K]]<br />
-[[Jacobi dn|Jacobi $\mathrm{dn}$]]</center></td>
-<td><center>[[File:jacobidsthumb.png|45px|link=Jacobi ds]]<br />
-[[Jacobi ds|Jacobi $\mathrm{ds}$]]</center></td>
-<td><center>[[File:jacobincthumb.png|45px|link=Jacobi nc]]<br />
-[[Jacobi nc|Jacobi $\mathrm{nc}$]]</center></td>
-<td><center>[[File:jacobindthumb.png|45px|link=Jacobi nd]]<br />
-[[Jacobi nd|Jacobi $\mathrm{nd}$]]</center></td>
-<td><center>[[File:jacobindthumb.png|45px|link=Jacobi ns]]<br />
-[[Jacobi ns|Jacobi $\mathrm{ns}$]]</center></td>
-<td><center>[[File:jacobiscthumb.png|45px|link=Jacobi sc]]<br />
-[[Jacobi sc|Jacobi $\mathrm{sc}$]]</center></td>
-</tr>
-<tr>
-<td><center>[[File:jacobisdthumb.png|45px|link=Jacobi sd]]<br />
-[[Jacobi sd|Jacobi $\mathrm{sd}$]]</center></td>
-<td><center>[[File:jacobisnthumb.png|45px|link=Jacobi sn]]<br />
-[[Jacobi sn|Jacobi $\mathrm{sn}$]]</center></td>
-<td><center>[[File:modularlambdathumb.png|45px|link=Modular lambda]]<br />
-[[Modular lambda|Modular $\lambda$]]</center></td>
-<td><center>[[File:weierstrasspthumb.png|45px|link=Weierstrass P]]<br />
-[[Weierstrass P|Weierstrass $P$]]</center></td>
-</tr>
-</table>
 [[Jacobi elliptic functions]]<br />
 [[Weierstrass sigma | Weierstrass $\sigma$]]<br />
+==Error function and friends==
+<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 />
+{{:Error functions footer}}
+==Figurate numbers==
+[[Tetrahedral numbers]]<br />
+[[Pyramidal numbers]] <br />
+[[Cubic numbers]]<br />
+[[Octahedral numbers]]<br />
+{{:Polygonal numbers footer}}
+[[Enneadecagonal numbers]]<br />
+[[Heptadecagonal numbers]]<br />
+[[Hexadecagonal numbers]]<br />
+[[Icosagonal numbers]]<br />
+[[Icosidigonal numbers]]<br />
+[[Icositetragonal numbers]]<br />
+[[Icositrigonal numbers]]<br />
+[[Icosihenagonal numbers]]<br />
+[[Myriagonal numbers]]<br />
+[[Octadecagonal numbers]]<br />
+[[Rectangular numbers]]<br />
+==Gamma function and friends==
+<div class="grid">
+<div class="grid"><center>[[Binomial coefficient|<randomimagebycategory categories="binomialglyph" width="45" />]]<br />
+[[Binomial coefficient]]</center></div>
+<div class="grid"><center>[[Barnes G|<randomimagebycategory categories="barnesgglyph" width="45" />]]<br />
+[[Barnes G|Barnes $G$]]</center></div>
+<div class="grid"><center>[[File:betathumb.png|45px|link=Beta]]<br />
+[[Beta]]</center></div>
+<div class="grid"><center>[[File:digammathumb.png|45px|link=Digamma]]<br />
+[[Digamma]]</center></div>
+<div class="grid"><center>[[File:factorialthumb.png|45px|link=Factorial]]<br />
+[[Factorial]]</center></div>
+<div class="grid"><center>[[File:hyperfactorialthumb.png|45px|link=Hyperfactorial]]<br />
+[[Hyperfactorial]]</center></div>
+<div class="grid"><center>[[File:gammathumb.png|45px|link=Gamma]]<br />
+[[Gamma]]</center></div>
+<div class="grid"><center>[[File:hadamardgammathumb.png|45px|link=Hadamard gamma]]<br />
+[[Hadamard gamma]]</center></div>
+<div class="grid"><center>[[File:reciprocalgammathumb.png|45px|link=Reciprocal gamma]]<br />
+[[Reciprocal gamma]]</center></div>
+<div class="grid"><center>[[File:digammathumb.png|45px|link=Polygamma]]<br />
+[[Polygamma]]</center></div>
+<div class="grid"><center>[[File:Incompletegammathumb.png|45px|link=Upper incomplete gamma]]<br />
+[[Upper incomplete gamma]]</center></div>
+<div class="grid"><center>[[File:pochhammerthumb.png|45px|link=Pochhammer]]<br />
+[[Pochhammer]]</center></div>
+</div>
+[[Falling factorial]]<br />
+[[Incomplete beta function]]<br />
+[[Loggamma]]<br />
+[[Lower incomplete gamma]]<br />
+[[Multiple gamma function]]<br />
+[[Primorial]]<br />
+[[Rising factorial]]<br />
+[[Trigamma]]<br />
+==Hypergeometric series==
+{{:Hypergeometric functions footer}}
+[[Hypergeometric 0F3]]<br />
+[[Hypergeometric 1F1]]<br />
+[[Hypergeometric 1F2]]<br />
+[[Hypergeometric 2F0]]<br />
+[[Hypergeometric 2F3]]<br />
+[[Hypergeometric 3F2]]<br />
+[[Hypergeometric 4F1]]<br />
+[[Lauricella–Saran functions]]<br />
+==*-integral functions==
+{{:*-integral functions footer}}
+[[Böhmer C]]<br />
+[[Böhmer S]]<br />
+[[Entire exponential integral]]<br />
+==Jacobi theta functions==
+{{:Jacobi theta footer}}
+==Logarithm function and friends==
+{{:Logarithm and friends footer}}
+[[Logarithm base a]]<br />
+[[Logarithm (multivalued)]]<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 />
+===Matrix hypergeometric functions===
+[[Matrix hypergeometric pFq]]<br />
+[[Matrix hypergeometric 0F3]]<br />
+[[Matrix hypergeometric 1F1]]<br />
+[[Matrix hypergeometric 1F2]]<br />
+[[Matrix hypergeometric 2F0]]<br />
+[[Matrix hypergeometric 2F3]]<br />
+[[Matrix hypergeometric 3F2]]<br />
+[[Matrix hypergeometric 4F1]]<br />
+==Operators==
+[[Alexander operator]]<br />
+[[Bernardi operator]]<br />
+[[Fourier transform]]<br />
+[[Laplace transform]]<br />
+[[Libera operator]]<br />
+[[Mellin transform]]<br />
+[[Radon transform]]<br />
+[[Two-dimensional Laplace transform]]<br />
 ==Polynomials==
-<table border="0">
+<div class="grid">
-<tr>
+<div class="grid2"><center>[[File:abelpolynomialthumb.png|45px|link=Abel p]]<br />
-<td><center>[[File:abelpolynomialthumb.png|45px|link=Abel p]]<br />
+[[Abel p|Abel $p$]]</center></div>
-[[Abel p|Abel $p$]]</center></td>
+</div>
-<td><center>[[File:associatedlaguerrelthumb.png|45px|link=Associated Laguerre L]]<br />
+[[Bernstein B]]<br />
-[[Associated Laguerre L]]</center></td>
-<td><center>[[File:batemanpolynomialthumb.png|45px|link=Bateman]]<br />
+==Probability distributions==
-[[Bateman Z | Bateman $Z$]]</center></td>
+===Continuous distributions===
-<td><center>[[File:bernoullibthumb.png|45px|link=Bernoulli B]]<br />
+====Probability density funcitons====
-[[Bernoulli B|Bernoulli $B$]]</center></td>
+[[Arcsin pdf]]<br />
-<td><center>[[File:besselpolynomialsthumb.png|45px|link=Bessel polynomial]]<br />
+[[Beta pdf]]<br />
-[[Bessel polynomial|Bessel poly]]</center></td>
+[[Cauchy pdf]]<br />
-<td><center>[[File:Chebyshevuthumb.png|45px|link=Chebyshev U]]<br />
+[[Chi-squared pdf]]<br />
-[[Chebyshev U]]</center></td>
+[[Exponential pdf]]<br />
-</tr>
+[[F pdf]]<br />
-<tr>
+[[Gamma pdf]]<br />
-<td><center>[[File:Chebyshevtthumb.png|45px|link=Chebyshev T]]<br />
+[[Laplace pdf]]<br />
-[[Chebyshev T]]</center></td>
+[[Log-normal pdf]]<br />
-<td><center>[[File:dicksonthumb.png|45px|link=Dickson polynomial]]<br />
+[[Normal pdf]]<br />
-[[Dickson polynomial|Dickson]]</center></td>
+[[Pareto pdf]]<br />
-<td><center>[[File:eulerethumb.png|45px|link=Euler E]]<br />
+[[Student's t pdf]]<br />
-[[Euler E|Euler $E$]]</center></td>
+[[Continuous uniform pdf]]<br />
-<td><center>[[File:fibonaccithumb.png|45px|link=Fibonacci]]<br />
+[[Weibull pdf]]<br />
-[[Fibonacci]]</center></td>
-<td><center>[[File:gegenbauercthumb.png|45px|link=Gegenbauer C]]<br />
+====Cumulative Density Functions====
-[[Gegenbauer C|Gegenbauer $C$]]</center></td>
+[[Arcsin cdf]]<br />
-<td><center>[[File:Hermitephysthumb.png|45px|link=Hermite (physicist)]]<br />
+[[Beta cdf]]<br />
-[[Hermite (physicist)|Hermite (phys)]]</center></td>
+[[Cauchy cdf]]<br />
-</tr>
+[[Chi-squared cdf]]<br />
-<tr>
+[[Exponential cdf]]<br />
-<td><center>[[File:Hermiteprobthumb.png|45px|link=Hermite (probabilist)]]<br />
+[[F cdf]]<br />
-[[Hermite (probabilist)|Hermite (prob)]]</center></td>
+[[Gamma cdf]]<br />
-<td><center>[[File:Jacobipthumb.png|45px|link=Jacobi P]]<br />
+[[Laplace cdf]]<br />
-[[Jacobi P|Jacobi $P$]]</center></td>
+[[Log-normal cdf]]<br />
-<td><center>[[File:laguerrelthumb.png|45px|link=Laguerre L]]<br />
+[[Normal cdf]]<br />
-[[Laguerre L|Laguerre $L$]]</center></td>
+[[Pareto cdf]]<br />
-<td><center>[[File:legendrepthumb.png|45px|link=Legendre P]]<br />
+[[Student's t cdf]]<br />
-[[Legendre P|Legendre $P$]]</center></td>
+[[Continuous uniform cdf]]<br />
-<td><center>[[File:lucasthumb.png|45px|link=Lucas]]<br />
+[[Weibull cdf]]<br />
-[[Lucas]]</center></td>
-</tr>
+===Discrete distributions===
-</table>
+[[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}}
+[[Adomian polynomials]]<br />
+[[Angelescu polynomials]]<br />
 [[Bell polynomial]]<br />
+[[Boole polynomials]]<br />
 [[Charlier polynomial]]<br />
+[[Cyclotomic polynomials]]<br />
+[[Denisyuk polynomials]]<br />
+[[Eberlein polynomials]]<br />
+[[Hahn polynomial]]<br />
+[[Humbert polynomials]]<br />
 [[q-Hermite polynomial | $q$-Hermite polynomial]]<br />
 [[Krawtchouk polynomial]]<br />
 [[Lagrange polynomial]]<br />
+[[Lidstone polynomial]]<br />
 [[Lommel polynomial]]<br />
+[[Macdonald polynomials]]<br />
 [[Mahler polynomial]]<br />
+[[Meixner polynomial]]<br />
+[[Meixner-Pollaczek polynomial]]<br />
 [[Mott polynomial]]<br />
+[[Narumi polynomials]]<br />
+[[Neumann polynomial]]<br />
+[[Padovan polynomials]]<br />
+[[Peters polynomials]]<br />
 [[Pidduck polynomial]]<br />
+[[Pincherle polynomials]]<br />
+[[Sister Celine's polynomials]]<br />
 [[Spread polynomial]]<br />
 [[Touchard polynomial]]<br />
 [[Rook polynomial]]<br />
+[[Stirling polynomial]]<br />
-==Trigonometric functions==
+==$q$-calculus==
-[[Böhmer integral]]<br />
+[[Basic Fourier series]]<br />
+[[Euler-Jackson q-difference operator|Euler-Jackson $q$-difference operator]]<br />
+[[q-analog|$q$-analog]]<br />
+[[q-Bernoulli numbers | $q$-Bernoulli numbers]]<br />
+[[q-derivative | $q$-derivative]]<br />
+[[q-numbers|$q$-numbers]]<br />
+[[Symmetric q-numbers|Symmetric $q$-numbers]]<br />
-===Standard trigonometric functions===
+===$q$-special functions===
-<table border="0">
+[[Basic hypergeometric phi|Basic hypergeometric $\phi$]]<br />
-<tr>
+[[Basic hypergeometric series psi|Basic hypergeometric series $\psi$]]<br />
-<td><center>[[File:Sinethumb.png|45px|link=Sine]]<br />
+[[Continuous q-Hermite polynomial | Continuous $q$-Hermite polynomials]]<br />
-[[Sine]]</center></td>
+[[Elliptic gamma function]]<br />
-<td><center>[[File:Cosinethumb.png|45px|link=Cosine]]<br />
+[[Exton q-exponential|Exton $q$-exponential]]<br />
-[[Cosine]]</center></td>
+[[Generalized q-Bessel | Generalized $q$-Bessel]]<br />
-<td><center>[[File:Tangentthumb.png|45px|link=Tangent]]<br />
+[[Hahn-Exton q-Bessel | Hahn-Exton $q$-Bessel]] <br />
-[[Tangent]]</center></td>
+[[Gosper q-sine | Gosper $q$-sine]]<br />
-<td><center>[[File:Cosecantthumb.png|45px|link=Cosecant]]<br />
+[[Jackson q-Bessel (1) | Jackson $q$-Bessel (1)]]<br />
-[[Cosecant]]</center></td>
+[[Jackson q-Bessel (2) | Jackson $q$-Bessel (2)]]<br />
-<td><center>[[File:Secantthumb.png|45px|link=Secant]]<br />
+[[Jackson q-Bessel (3) | Jackson $q$-Bessel (3)]]<br />
-[[Secant]]</center></td>
+[[LLT polynomials]]<br />
-<td><center>[[File:Cotangentthumb.png|45px|link=Cotangent]]<br />
+[[q-Bessel functions | $q$-Bessel function]]<br />
-[[Cotangent]]</center></td>
+[[q-Beta function | $q$-Beta function]]<br />
-</tr>
+[[q-cos | $q$-$\cos$]]<br />
-</table>
+[[q-Cos | $q$-$\mathrm{Cos}$]]<br />
+[[q-Binomial coefficient | $q$-Binomial coefficient]]<br />
+[[q-Dirichlet series | $q$-Dirichlet series]]<br />
+[[q-exponential e sub q | $q$-exponential $e_q$]]<br />
+[[q-exponential e sub 1/q | $q$-exponential $e_{\frac{1}{q}}$]]<br />
+[[q-exponential E sub q| $q$-exponential $E_q$]]<br />
+[[q-exponential E sub 1/q | $q$-exponential $E_{\frac{1}{q}}$]]<br />
+[[q-Fibonacci polynomials | $q$-Fibonacci polynomials]]<br />
+[[q-Gamma function]] <br />
+[[q-Gaussian distribution | $q$-Gaussian distribution]]<br />
+[[q-Hermite polynomial | $q$-Hermite polynomial]]<br />
+[[q-Hurwitz zeta|$q$-Hurwitz zeta]]<br />
+[[q-Pochhammer | $q$-Pochhammer symbol]]<br />
+[[q-Pollaczek polynomial]]<br />
+[[q-Polygamma function | $q$-Polygamma function]]<br />
+[[q-shifted factorial | $q$-shifted factorial]]<br />
+[[q-sin | $q$-$\sin$]]<br />
+[[q-Sin | $q$-$\mathrm{Sin}$]]<br />
+[[q-Theta function | $q$-Theta function]]<br />
+[[q-ultraspherical polynomial|$q$-ultraspherical polynomial]]<br />
+[[q-zeta|$q$-zeta]]<br />
+[[Second q-shifted factorial|Second $q$-shifted factorial]]<br />
+[[Tsallis q-exponential|Tsallis $q$-exponential]]<br />
-===Inverse trigonometric functions===
+{{:Q-calculus footer}}
-<table border="0">
-<tr>
-<td><center>[[File:Arcsinthumb.png|45px|link=Arcsin]]<br />
-[[Arcsin]]</center></td>
-<td><center>[[File:Arccosthumb.png|45px|link=Arccos]]<br />
-[[Arccos]]</center></td>
-<td><center>[[File:Arctanthumb.png|45px|link=Arctan]]<br />
-[[Arctan]]</center></td>
-<td><center>[[File:Arccscthumb.png|45px|link=Arccsc]]<br />
-[[Arccsc]]</center></td>
-<td><center>[[File:Arcsecthumb.png|45px|link=Arcsec]]<br />
-[[Arcsec]]</center></td>
-<td><center>[[File:Arccotthumb.png|45px|link=Arccot]]<br />
-[[Arccot]]</center></td>
-</tr>
-</table>
-===Hyperbolic trigonometric functions===
+==Special functions in number theory==
-<table border="0">
+{{:Number theory functions footer}}
-<tr>
-<td><center>[[File:Sinhthumb.png|45px|link=Sinh]]<br />
-[[Sinh]]</center></td>
-<td><center>[[File:Coshthumb.png|45px|link=Cosh]]<br />
-[[Cosh]]</center></td>
-<td><center>[[File:Tanhthumb.png|45px|link=Tanh]]<br />
-[[Tanh]]</center></td>
-<td><center>[[File:Cschthumb.png|45px|link=Csch]]<br />
-[[Csch]]</center></td>
-<td><center>[[File:Sechthumb.png|45px|link=Sech]]<br />
-[[Sech]]</center></td>
-<td><center>[[File:Coththumb.png|45px|link=Coth]]<br />
-[[Coth]]</center></td>
-</tr>
-</table>
-===Inverse hyperbolic trigonometric functions===
+[[Dirichlet L-function | Dirichlet $L$-function]]<br />
-<table border="0">
+[[p-adic L function|$p$-adic $L$ function]]<br />
-<tr>
+[[Ramanujan's sum]]<br />
-<td><center>[[File:Arcsinhthumb.png|45px|link=Arcsinh]]<br />
+[[Riemann prime counting]]<br />
-[[Arcsinh]]</center></td>
-<td><center>[[File:Arccoshthumb.png|45px|link=Arccosh]]<br />
-[[Arccosh]]</center></td>
-<td><center>[[File:Arctanhthumb.png|45px|link=Arctanh]]<br />
-[[Arctanh]]</center></td>
-<td><center>[[File:Arccschthumb.png|45px|link=Arccsch]]<br />
-[[Arccsch]]</center></td>
-<td><center>[[File:Arcsechthumb.png|45px|link=Arcsech]]<br />
-[[Arcsech]]</center></td>
-<td><center>[[File:Arccoththumb.png|45px|link=Arccoth]]<br />
-[[Arccoth]]</center></td>
-</tr>
-</table>
-===$*$-c functions===
+==Special sequences==
-<table>
+Sequences are merely functions whose domain is a subset of $\mathbb{Z}$. <br />
-<tr>
+[[Amicable numbers]] <br />
-<td><center>[[File:Sincthumb.png|45px|link=Sinc]]<br />
+[[Associated Stirling numbers of the second kind]]<br />
-[[Sinc]]</center></td>
+[[Bell numbers]]<br />
-<td><center>[[File:tancthumb.png|45px|link=Tanc]]<br />
+[[Bernoulli numbers]] ($B_0 = 1, B_1 = \pm \frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = −\frac{1}{30}, B_5 = 0, B_6 = \frac{1}{42}, B_7 = 0, B_8 = −\frac{1}{30}$) <br />
-[[Tanc]]</center></td>
+[[Betrothed numbers]]<br />
-</tr>
+[[Carmichael numbers]] <br />
-</table>
+[[Euler numbers]] <br />
+[[Erdős–Nicolas numbers]]<br />
+[[Farey sequence]]<br />
+[[Fermat numbers]]<br />
+[[Fibonacci numbers]]<br />
+[[Genocchi numbers]]<br />
+[[Harmonic number]]<br />
+[[Idoneal number]]<br />
+[[Jacobsthal numbers]]<br />
+[[Signed Lah numbers]]<br />
+[[Unsigned Lah numbers]]<br />
+[[Lucas numbers]]<br />
+[[Lychrel numbers]]<br />
+[[Pell numbers]]<br />
+[[Mersenne numbers]]<br />
+[[Nielsen-Ramanujan sequence]]<br />
+[[Perfect numbers]]<br />
+[[Powerful numbers]]<br />
+[[Pronic numbers]]<br />
+[[q-Bernoulli numbers | $q$-Bernoulli numbers]]<br />
+[[Reduced Stirling numbers of the second kind]]<br />
+[[Stieltjes constants]]<br />
+[[Stirling numbers of the first kind]]<br />
+[[Stirling numbers of the second kind]]<br />
+[[Sylvester's sequence]]<br />
+[[Thue-Morse sequence]]<br />
-===$*$-integral functions===
+==Trigonometric functions==
-<table border="0">
+[[Arcsin (multivalued)]] <br />
-<tr>
+[[Arccos (multivalued)]] <br />
-<td><center>[[File:Cosintegralthumb.png|45px|link=Cosine integral]]<br />
+[[Arctan (multivalued]] <br />
-[[Cosine integral]]</center></td>
+[[Arccsc (multivalued)]] <br />
-<td><center>[[File:Coshintegralthumb.png|45px|link=Hyperbolic cosine integral]]<br />
+[[Arcsec (multivalued)]] <br />
-[[Hyperbolic cosine integral|Cosh integral]]</center></td>
+[[Arccot (multivalued)]] <br />
-<td><center>[[File:Fresnelcthumb.png|45px|link=Fresnel C]]<br />
-[[Fresnel C | Fresnel $C$]]</center></td>
-<td><center>[[File:eithumb.png|45px|link=Exponential integral]]<br /> [[Exponential integral]]</center></td>
-<td><center>[[File:Fresnelsthumb.png|45px|link=Fresnel S]]<br />
-[[Fresnel S]]</center></td>
-<td><center>[[File:Sinintegralthumb.png|45px|link=Sine integral]]<br />
-[[Sine integral]]</center></td>
-<td><center>[[File:Sinhintegralthumb.png|45px|link=Hyperbolic sine integral]]<br />
-[[Hyperbolic sine integral|Sinh integral]]</center></td>
-</tr>
-</table>
-==Special functions in number theory==
+{{:Trigonometric functions footer}}
-<table border="0">
+{{:Inverse trigonometric functions footer}}
-<tr>
+{{:Hyperbolic trigonometric functions footer}}
-<td><center>[[File:eulerphithumb.png|45px|link=Euler totient]]<br />
+{{:Inverse hyperbolic trigonometric functions footer}}
-[[Euler totient|Euler totient]]</center></td>
+{{:Depreciated trigonometric functions footer}}
-<td><center>[[File:mangoldtthumb.png|45px|link=Mangoldt]]<br />
-[[Mangoldt]]</center></td>
-<td><center>[[File:mertensthumb.png|45px|link=Mertens]]<br />
-[[Mertens]]</center></td>
-<td><center>[[File:möbiusthumb.png|45px|link=Möbius]]<br />
-[[Möbius]]</center></td>
-<td><center>[[File:liouvillelambdathumb.png|45px|link=Liouville lambda]]<br />
-[[Liouville lambda|Liouville $\lambda$]]</center></td>
-<td><center>[[File:primepithumb.png|45px|link=Prime counting]]<br />
-[[Prime counting]]</center></td>
-</tr>
-<tr>
-<td><center>[[File:riemannzetathumb.png|45px|link=Riemann zeta]]<br />
-[[Riemann zeta]]</center></td>
-<td><center>[[File:divisorsumthumb.png|45px|link=Sum of divisors]]<br />
-[[Sum of divisors]]</center></td>
-</tr>
-</table>
-[[Chebyshev function]]<br />
+===Depreciated trigonometric functions===
-[[Dirichlet L-function | Dirichlet $L$-function]]<br />
+[[Arccosvercosine]]<br />
-[[Greatest prime factor]]<br />
+[[Arccoversine]]<br />
-[[Ramanujan's sum]]<br />
+[[Archavercosine]]<br />
-[[Riemann prime counting]]<br />
+[[Archaversine]]<br />
+[[Arcvercosine]]<br />
+[[Arcversine]]<br />
+[[Hacovercosine]]<br />
+[[Hacoversine]]<br />
+[[Havercosine]]<br />
+[[Haversine]]<br />
+[[Inverse tangent integral]]<br />
+[[Vercosine]]<br />
 ==Zeta functions==
 A [http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/directoryofzetafunctions.htm directory] of zeta functions.
-<table border="0">
+<div class="grid">
-<tr>
+<div class="grid"><center>[[File:hurwitzzetathumb.png|45px|link=Hurwitz zeta]]<br />
-<td><center>[[File:hurwitzzetathumb.png|45px|link=Hurwitz zeta]]<br />
+[[Hurwitz zeta]]</center></div>
-[[Hurwitz zeta]]</center></td>
+<div class="grid"><center>[[File:jacobizetathumb.png|45px|link=Jacobi zeta]]<br />
-<td><center>[[File:jacobizetathumb.png|45px|link=Jacobi zeta]]<br />
+[[Jacobi zeta]]</center></div>
-[[Jacobi zeta]]</center></td>
+<div class="grid"><center>[[File:primezetapthumb.png|45px|link=Prime zeta P]]<br />
-<td><center>[[File:primezetapthumb.png|45px|link=Prime zeta P]]<br />
+[[Prime zeta P|Prime zeta $P$]]</center></div>
-[[Prime zeta P|Prime zeta $P$]]</center></td>
+<div class="grid"><center>[[File:riemannzetathumb.png|45px|link=Riemann zeta]]<br />
-<td><center>[[File:riemannzetathumb.png|45px|link=Riemann zeta]]<br />
+[[Riemann zeta]]</center></div>
-[[Riemann zeta]]</center></td>
+<div class="grid"><center>[[File:weierstrasszetathumb.png|45px|link=Weierstrass zeta]]<br />
-<td><center>[[File:weierstrasszetathumb.png|45px|link=Weierstrass zeta]]<br />
+[[Weierstrass zeta]]</center></div>
-[[Weierstrass zeta]]</center></td>
+</div>
-</tr>
-</table>
 [[Airy zeta function]]<br />
@@ Line 396: / Line 510: @@
 [[Artin-Mazur zeta function]]<br />
 [[Barnes zeta function]]<br />
+[[Bessel zeta function]]<br />
 [[Beurling zeta function]]<br />
 [[Cotangent zeta function]]<br />
 [[Dedekind zeta function]]<br />
 [[Epstein zeta function]]<br />
+[[Fibonacci zeta function]]<br />
 [[Goss zeta function]]<br />
 [[Hasse-Weil zeta function]]<br />
@@ Line 413: / Line 529: @@
 [[Multiple zeta function]]<br />
 [[p-adic zeta function | $p$-adic zeta function]]<br />
+[[q-Hurwitz zeta|$q$-Hurwitz zeta]]<br />
+[[q-zeta|$q$-zeta]]<br />
 [[Reciprocal zeta function]]<br />
 [[Ruelle zeta function]]<br />
@@ Line 420: / Line 538: @@
 [[Shintani zeta function]]<br />
 [[Witten zeta function]]<br />
+[[Z function]]<br />
-==$q$-Calculus==
+==Arithmetic Operations==
-<table border="0">
+[[Addition]] <br />
-<tr>
+[[Division]] <br />
-<td><center>[[File:qbinomialthumb.png|45px|link=q-Binomial]]<br />
+[[Exponentiation]]<br />
-[[q-Binomial|$q$-Binomial]]</center></td>
+[[Hyperoperation]]<br />
-<td><center>[[File:qgammathumb.png|45px|link=q-Gamma]]<br />
+[[Multiplication]] <br />
-[[q-Gamma|$q$-Gamma]]</center></td>
+[[Subtraction]] <br />
-<td><center>[[File:qfactorialthumb.png|45px|link=q-Factorial]]<br />
+[[Tetration]]<br />
-[[q-factorial|$q$-factorial]]</center></td>
-<td><center>[[File:Eulerqphi.png|45px|link=Euler phi]]<br />
-[[Euler phi|Euler $\phi$]]</center></td>
-</tr>
-</table>
-[[Basic Fourier series]]<br />
-[[Basic hypergeometric series phi|Basic hypergeometric series $\phi$]]<br />
-[[Basic hypergeometric series psi|Basic hypergeometric series $\psi$]]<br />
-[[Continuous q-Hermite | Continuous $q$-Hermite polynomials]]<br />
-[[Hahn-Exton q-Bessel | Hahn-Exton $q$-Bessel]] <br />
-[[Gosper q-sine | Gosper $q$-sine]]<br />
-[[Jackson q-Bessel (1) | Jackson $q$-Bessel (1)]]<br />
-[[Jackson q-Bessel (2) | Jackson $q$-Bessel (2)]]<br />
-[[Jackson q-Bessel (3) | Jackson $q$-Bessel (3)]]<br />
-[[q-analog|$q$-analog]]<br />
-[[q-Bernoulli numbers | $q$-Bernoulli numbers]]<br />
-[[q-Bessel functions | $q$-Bessel function]]<br />
-[[q-Beta function | $q$-Beta function]]<br />
-[[q-cos | $q$-$\cos$]]<br />
-[[q-Cos | $q$-$\mathrm{Cos}$]]<br />
-[[q-derivative | $q$-derivative]]<br />
-[[q-exponential e| $q$-exponential e]]<br />
-[[q-exponential E| $q$-exponential E]]<br />
-[[q-Fibonacci polynomials | $q$-Fibonacci polynomials]]<br />
-[[q-Gaussian distribution | $q$-Gaussian distribution]]<br />
-[[q-Hermite polynomial | $q$-Hermite polynomial]]<br />
-[[q-Hypergeometric function | $q$-Hypergeometric function]]<br />
-[[q-numbers|$q$-numbers]]<br />
-[[q-Pochhammer | $q$-Pochhammer symbol]]<br />
-[[q-Polygamma function | $q$-Polygamma function]]<br />
-[[q-shifted factorial | $q$-shifted factorial]]<br />
-[[q-sin | $q$-$\sin$]]<br />
-[[q-Sin | $q$-$\mathrm{Sin}$]]<br />
-[[q-Theta function | $q$-Theta function]]<br />
-[[Tsallis q-exponential|Tsallis $q$-exponential]]<br />
-=Special sequences=
-[[Bell numbers]]<br />
-[[Bernoulli numbers]] ($B_0 = 1, B_1 = \pm \frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = −\frac{1}{30}, B_5 = 0, B_6 = \frac{1}{42}, B_7 = 0, B_8 = −\frac{1}{30}$) <br />
-[[Euler numbers]] <br />
-[[Genocchi numbers]]<br />
-[[Harmonic number]]<br />
-[[Lah numbers]]<br />
-[[Mersenne numbers]]<br />
-[[Stirling numbers of the first kind]]<br />
-[[Stirling numbers of the second kind]]<br />
-==Figurate numbers==
-[[Square number]]<br />
-[[Triangular numbers]]<br />
-[[Tetrahedral numbers]]<br />
 =Special constants=
-[[Apéry's constant]] ($\zeta(3)=1.2020569031595942854 \ldots$)<br />
+[[Algebraic]] <br />
-[[Catalan constant]] ($0.915965594177219015054603514932384110774 \ldots$)<br />
+[[Transcendental]]
-[[Chaitin's constant]]<br />
+{| class="wikitable sortable"
-[[Champernowne constant]] ($0.12345678910111213141516171819202122232425\ldots$)<br />
+|-
-[[Conway's constant]] ($1.303577269034 \ldots$)<br />
+! Name !! Notation !! Value
-[[Copeland-Erdős constant]] ($0.23571113171923...$)<br />
+|-
-[[E | Euler's number]] ($e=2.71828182846 \ldots$)<br />
+| [[Apéry's constant]] || $\zeta(3)$  || $1.2020569031595942854 \ldots$
-[[Erdős-Borwein Constant]] ($E=1.606695152415291763\ldots$)<br />
+|-
-[[Euler-Mascheroni constant | Euler-Mascheroni constant]] ($\gamma=0.5772156649015328606065120900824024310421593359399235\ldots$)<br />
+| [[Artin constant]] || $C_{\mathrm{Artin}}$ || $0.3739558136\ldots$
-[[Feigenbaum constants]] ($4.669201609102990671853203820466201617258185577475768632745651 \ldots$)<br />
+|-
-[[Fransén–Robinson constant]] ($2.807770242028519365221501186557772932308085920930198291220054 \ldots$)<br />
+| [[Brun's constant]] || $B_2$ || $1.9021605831\ldots$
-[[Gauss' constant]]<br />
+|-
-[[Gelfond constant]] ($e^{\pi}=23.14069263277926900572908636794854738026610624260021199344504 \ldots$)<br />
+| [[Cahen's constant]] || $C$ || $0.64341054629\ldots$
-[[Gelfond–Schneider constant]] ($2^{\sqrt{2}}=2.665144142690225188650297249873139848274211313714659492835979 \ldots$)<br />
+|-
-[[Glaisher–Kinkelin constant]] ($A = 1.2824271291 \ldots$)<br />
+| [[Catalan constant]] || $G$ || $0.915965594177219015054603514932384110774 \ldots$
-[[Golden ratio]] ($\phi=1.618033988749894848204586834365638117720309179805762862135448 \ldots$)<br />
+|-
-[[Imaginary number]] ($i=\sqrt{-1}$)<br />
+| [[Chaitin's constant]] || ||
-[[Kaprekar's constant]]<br />
+|-
-[[Khinchin's constant]] ($K=2.685452001065306445309714835481795693820382293994462953051152 \ldots$)<br />
+| [[Champernowne constant]] || $C_{10}$ || $0.12345678910111213141516171819202122232425\ldots$
-[[Legendre's constant]] ($B=1$)<br />
+|-
-[[Lefschetz number]]<br />
+| [[Conway's constant]] || || $1.303577269034 \ldots$
-[[Lévy's constant]]<br />
+|-
-[[Mills' constant]] ($M=1.306377883863080690468614492602605712916784585156713644368053 \ldots$)<br />
+| [[Copeland-Erdős constant]] || || $0.23571113171923...$
-[[Pi]] ($\pi=3.141592653589793238462643383279502884197169399375105820974944 \ldots$)<br />
+|-
-[[q-Bernoulli numbers | $q$-Bernoulli numbers]]<br />
+| [[E | Euler's number]] || $e$ || $2.71828182846 \ldots$
-[[q-pi|$\pi_q$]]<br />
+|-
-[[Silver ratio]] ($\delta_s=2.4142135623730950488 \ldots$)<br />
+| [[Erdős-Borwein Constant]] || $E$ || $1.606695152415291763\ldots$
-[[Soldner's Constant]] ($\mu = 1.45136923488338105028396848589202744949\ldots$) <br />
+|-
+| [[Euler-Mascheroni constant | 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=
@@ Line 517: / Line 688: @@
 [[Irrational number]] $\mathbb{R} \setminus \mathbb{Q}$<br />
 [[Natural number]] $\mathbb{N}$<br />
+[[p-adic number]]<br />
 [[Prime number]] <br />
 [[Rational number]] $\mathbb{Q}$<br />
 [[Real number]] $\mathbb{R}$<br />
+[[Transcendental number]] <br />
 =Tools from calculus=
@@ Line 525: / Line 698: @@
 [[Absolute minimum]]<br />
 [[Analytic continuation]]<br />
+[[Arithmetic functions]]<br />
+[[Binomial series]]<br />
+[[Cell]]<br />
 [[Continuous]]<br />
 [[Derivative]]<br />
 [[Difference equation of hypergeometric type]]<br />
+[[Doubly periodic function]]<br />
+[[Entire]]<br />
+[[Fundamental pair of periods]]<br />
 [[Integration by parts]]<br />
+[[Integral from a to a]]<br />
 [[Laplace transform]]<br />
+[[Lattice generated by doubly periodic periods]]<br />
 [[Lebesgue integral]]<br />
 [[Local maximum]]<br />
 [[Local minimum]]<br />
+[[Periodic function]]<br />
+[[Period parallelogram]]<br />
+[[Polar coordinates]]<br />
+[[Product rule for derivatives]]<br />
+[[Quotient rule]]<br />
 [[Ratio_test | Ratio test]]<br />
 [[Riemann integral]]<br />
+[[Stirling formula]]<br />
 [[Rolle's theorem]]<br />
 [[Taylor series]]<br />
@@ Line 540: / Line 727: @@
 =Other=
-[http://dualaud.net/txt/rainvillespecialfunctions.php Solution manual to Rainville's "Special Functions"]<br />
+[[Functions named after Pafnuty Chebyshev]]<br />
+[[Functions named after Peter Gustav Lejeune Dirichlet]]<br />
+[[Functions named after Leonard Euler]]<br />
+[[Functions named after Carl Gustav Jacob Jacobi]]<br />
+[[Functions named after Bernhard Riemann]]<br />
+[[Functions named after Karl Weierstrass]]<br />
+=External Links=
+[http://authors.library.caltech.edu/43491/ Higher transcendental functions (Bateman project)]<br />
+[https://github.com/tomcuchta/rainvillesfsolutions Solution manual to Rainville's "Special Functions"]<br />
 [http://web.mst.edu/~lmhall/SPFNS/spfns.pdf "Special Functions" by Leon Hall]<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://www.people.fas.harvard.edu/~sfinch/ Mathematical Constants by Steven R. Finch]<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://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/index.htm Abramowitz and Stegun]<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 Combinatorcs/]<br />
+[http://algo.inria.fr/flajolet/Publications/books.html Phillipe Flajolet's "Analytic Combinatorics"]<br />
-[http://mpmath.googlecode.com/svn/gallery/gallery.html Gallery of mathematical functions]<br />
 [http://www.numbertheory.org/ntw/lecture_notes.html Online number theory lecture notes and teaching materials]<br />
-[http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]
+[http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]<br />
+[http://scipp.ucsc.edu/~haber/archives/physics116A10/arccoth.pdf "According to Abramowitz and Stegun" or arccoth needn't be uncouth]<br />

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

Difference between revisions of "Main Page"

Latest revision as of 13:31, 8 November 2024

Contents

Special functions

Bessel functions and friends

*-c functions

Continuous nowhere-differentiable functions and friends

Defined by a differential equation

Elliptic functions

Error function and friends

Figurate numbers

Gamma function and friends

Hypergeometric series

*-integral functions

Jacobi theta functions

Logarithm function and friends

Matrix functions

Matrix hypergeometric functions

Operators

Polynomials

Probability distributions

Continuous distributions

Probability density funcitons

Cumulative Density Functions

Discrete distributions

$q$-calculus

$q$-special functions

Special functions in number theory

Special sequences

Trigonometric functions

Depreciated trigonometric functions

Zeta functions

Arithmetic Operations

Special constants

Numbers

Tools from calculus

Other

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