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.
-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.
+<b><u>Registration</u></b>
+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=
-[[Airy functions]]<br />
+<div class="grid">
-[[Automorphic forms and functions]]<br />
+<div class="grid"><center>[[Ceiling|<randomimagebycategory categories="ceilingglyph" width="45" />]] <br />
-[[Bessel functions]]<br />
+[[Ceiling]]</center> </div>
-[[Beta function]]<br />
+<div class="grid"><center>[[File:dedekindetathumb.png|45px|link=Dedekind eta]]<br />
-[[Binomial function]]<br />
+[[Dedekind eta]]</center></div>
+<div class="grid"><center>[[File:dirichletetathumb.png|45px|link=Dirichlet eta]]<br />
+[[Dirichlet eta]]</center></div>
+<div class="grid"><center>[[File:dirichletbetathumb.png|45px|link=Dirichlet beta]]<br />
+[[Dirichlet beta]]</center></div>
+<div class="grid"><center>[[Exponential|<randomimagebycategory categories="exponentialglyph" width="45" />]]<br />
+[[Exponential]]</center></div>
+<div class="grid"><center>[[Floor|<randomimagebycategory categories="floorglyph" width="45" />]]<br />
+[[Floor]]</center></div>
+<div class="grid"><center>[[Identity function|<randomimagebycategory categories="identityglyph" width="45" />]]<br />
+[[Identity function|Identity]]</center></div>
+<div class="grid"><center>[[File:kleininvariantjthumb.png|45px|link=Klein invariant J]]<br />
+[[Klein invariant J|Klein invariant $J$]]</center></div>
+<div class="grid"><center>[[File:partitionthumb.png|45px|link=Partition]]<br />
+[[Partition]]</center></div>
+<div class="grid"><center>[[File:lambertwthumb.png|45px|link=Lambert W]]<br />
+[[Lambert W|Lambert $W$]]</center></div>
+<div class="grid"><center>[[File:planckradiationthumb.png|45px|link=Planck radiation]]<br />
+[[Planck radiation]]</center></div>
+<div class="grid"><center>[[File:ramanujantauthumb.png|45px|link=Ramanujan tau]]<br />
+[[Ramanujan tau|Ramanujan $\tau$]]</center></div>
+<div class="grid"><center>[[File:riemannxithumb.png|45px|link=Riemann xi]]<br />
+[[Riemann xi|Riemann $\xi$]]</center></div>
+<div class="grid"><center>[[Signum|<randomimagebycategory categories="Signumglyph" width="45" />]]<br />
+[[Signum]]</center></div>
+<div class="grid"><center>[[File:thomaethumb.png|45px|link=Thomae]]<br />
+[[Thomae]]</center></div>
+</div>
+[[Bickley-Naylor]]<br />
 [[Carlitz-Goss Gamma function]]<br />
-[[Clausen functions]]<br />
+[[Clausen cosine]]<br />
-[[Dawson function]]<br />
+[[Clausen sine]]<br />
-[[Dedekind eta function]]<br />
+[[Debye function]]<br />
-[[Digamma function]]<br />
+[[Dirichlet function]]<br />
-[[Dirichlet eta function]]<br />
+[[Discriminant]]<br />
+[[Distance to integers]]<br />
 [[Eisenstein series]]<br />
-[[Error function]]<br />
+[[Heaviside step function]]<br />
-[[Exponential function]]<br />
-[[Exponential integral]]<br />
-[[Factorial]]<br />
-[[Fresnel integral]]<br />
-[[Gamma function]]<br />
-[[Generalized_hypergeometric_function | Generalized hypergeometric function]] ${}_pF_q$ <br />
 [[Hermite function]]<br />
-[[Hypergeometric q-series | Hypergeometric $q$-series]]<br />
+[[K-function]]<br />
-[[Jacobi elliptic functions]]<br />
+[[Legendre chi]]<br />
-[[Jacobi theta functions]]<br />
-[[Klein's j-invariant]]<br />
-[[Lambert W function | Lambert $W$ function]]<br />
 [[Legendre function]]<br />
 [[Lerch transcendent]]<br />
-[[Logarithm]]<br />
+[[Log barrier]]<br />
-[[Logarithmic integral]] <br />
+[[MacRobert E]]<br />
-[[Mathieu function]] <br />
 [[Maass forms]] <br />
-[[Minkowski question mark function | Minkowski $?$ function]]<br />
 [[Meijer G-function]]<br />
-[[Modular forms and functions]]<br />
+[[Mittag-Leffler]]<br />
+[[Modular form]]<br />
 [[Mock modular forms]]<br />
 [[Mock theta functions]]<br />
+[[Nome]]<br />
 [[Painlevé transcendents]]<br />
-[[Polylogarithm]]<br />
+[[Polyexponential]]<br />
-[[Partition function]]<br />
+[[Ramanujan theta function]]<br />
-[[Pochhammer symbol]]<br />
+[[Riemann-Landau xi]]<br />
-[[Polygamma functions]]<br />
+[[Riemann Siegel theta function]]<br />
-[[Polylogarithms]]<br />
 [[Riemann theta function]]<br />
-[[Sum of positive divisors function]]<br />
+[[Sievert integral]]<br />
-[[Sinc function]]<br />
+[[Singular function]]<br />
-[[Struve functions]]<br />
 [[Theta functions]]<br />
-[[Trigonometric integral]]<br />
+[[Unit step function]]<br />
-[[Weierstrass elliptic functions]]<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}}
+[[Besicovitch functions]]<br />
+[[Bolzano function]]<br />
+[[Cellérier function]]<br />
+[[Katsuura function]]<br />
+[[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 />
+==Defined by a differential equation==
+[[Buchstab function]]<br />
+[[Dickman–de Bruijn function]]<br />
+[[Mathieu function]] <br />
+==Elliptic functions==
+[[Elliptic function|Elliptic functions (general overview and definitions)]]
+<div class="grid">
+<div class="grid"><center>[[File:ellipticethumb.png|45px|link=Elliptic E]]<br />
+[[Elliptic E|Elliptic $E$]]</center></div>
+<div class="grid"><center>[[File:elliptickthumb.png|45px|link=Elliptic K]]<br />
+[[Elliptic K|Elliptic $K$]]</center></div>
+<div class="grid"><center>[[File:ellipticthetathumb.png|45px|link=Elliptic theta]]<br />
+[[Elliptic theta|Elliptic $\theta$]]</center></div>
+<div class="grid"><center>[[File:modularlambdathumb.png|45px|link=Modular lambda]]<br />
+[[Modular lambda|Modular $\lambda$]]</center></div>
+<div class="grid"><center>[[File:weierstrasspthumb.png|45px|link=Weierstrass elliptic]]<br />
+[[Weierstrass elliptic|Weierstrass elliptic]]</center></div>
+</div>
+{{:Jacobi elliptic functions footer}}
+[[Incomplete Elliptic E]]<br />
+[[Incomplete Elliptic K]]<br />
+[[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==
-[[Abel polynomial]]<br />
+<div class="grid">
+<div class="grid2"><center>[[File:abelpolynomialthumb.png|45px|link=Abel p]]<br />
+[[Abel p|Abel $p$]]</center></div>
+</div>
+[[Bernstein B]]<br />
+==Probability distributions==
+===Continuous distributions===
+====Probability density funcitons====
+[[Arcsin pdf]]<br />
+[[Beta pdf]]<br />
+[[Cauchy pdf]]<br />
+[[Chi-squared pdf]]<br />
+[[Exponential pdf]]<br />
+[[F pdf]]<br />
+[[Gamma pdf]]<br />
+[[Laplace pdf]]<br />
+[[Log-normal pdf]]<br />
+[[Normal pdf]]<br />
+[[Pareto pdf]]<br />
+[[Student's t pdf]]<br />
+[[Continuous uniform pdf]]<br />
+[[Weibull pdf]]<br />
+====Cumulative Density Functions====
+[[Arcsin cdf]]<br />
+[[Beta cdf]]<br />
+[[Cauchy cdf]]<br />
+[[Chi-squared cdf]]<br />
+[[Exponential cdf]]<br />
+[[F cdf]]<br />
+[[Gamma cdf]]<br />
+[[Laplace cdf]]<br />
+[[Log-normal cdf]]<br />
+[[Normal cdf]]<br />
+[[Pareto cdf]]<br />
+[[Student's t cdf]]<br />
+[[Continuous uniform cdf]]<br />
+[[Weibull cdf]]<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}}
+[[Adomian polynomials]]<br />
+[[Angelescu polynomials]]<br />
 [[Bell polynomial]]<br />
-[[Bernoulli polynomial]]<br />
+[[Boole polynomials]]<br />
-[[Chebyshev polynomial]]<br />
+[[Charlier polynomial]]<br />
-[[Dickson polynomial]]<br />
+[[Cyclotomic polynomials]]<br />
-[[Euler polynomial]]<br />
+[[Denisyuk polynomials]]<br />
-[[Fibonacci polynomial]]<br />
+[[Eberlein polynomials]]<br />
-[[Hermite polynomial]]<br />
+[[Hahn polynomial]]<br />
+[[Humbert polynomials]]<br />
 [[q-Hermite polynomial | $q$-Hermite polynomial]]<br />
+[[Krawtchouk polynomial]]<br />
 [[Lagrange polynomial]]<br />
-[[Laguerre polynomial]]<br />
+[[Lidstone polynomial]]<br />
-[[Legendre polynomial]]<br />
+[[Lommel polynomial]]<br />
-[[Lucas 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 />
+==$q$-calculus==
+[[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 />
+===$q$-special functions===
+[[Basic hypergeometric phi|Basic hypergeometric $\phi$]]<br />
+[[Basic hypergeometric series psi|Basic hypergeometric series $\psi$]]<br />
+[[Continuous q-Hermite polynomial | Continuous $q$-Hermite polynomials]]<br />
+[[Elliptic gamma function]]<br />
+[[Exton q-exponential|Exton $q$-exponential]]<br />
+[[Generalized q-Bessel | Generalized $q$-Bessel]]<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 />
+[[LLT polynomials]]<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-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 />
-==Trigonometric functions==
+{{:Q-calculus footer}}
-[[Cosine function]]<br />
-[[Sine function]]<br />
-[[Tangent function]]<br />
 ==Special functions in number theory==
-[[Chebyshev function]]<br />
+{{:Number theory functions footer}}
 [[Dirichlet L-function | Dirichlet $L$-function]]<br />
-[[Liouville function]]<br />
+[[p-adic L function|$p$-adic $L$ function]]<br />
-[[Mangoldt function]]<br />
-[[Mertens function]]<br />
-[[Möbius function]]<br />
-[[Prime counting function]]<br />
 [[Ramanujan's sum]]<br />
-[[Riemann zeta function]]<br />
+[[Riemann prime counting]]<br />
-[[Totient function]]<br />
+==Special sequences==
+Sequences are merely functions whose domain is a subset of $\mathbb{Z}$. <br />
+[[Amicable numbers]] <br />
+[[Associated Stirling numbers of the second kind]]<br />
+[[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 />
+[[Betrothed numbers]]<br />
+[[Carmichael numbers]] <br />
+[[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 />
+==Trigonometric functions==
+[[Arcsin (multivalued)]] <br />
+[[Arccos (multivalued)]] <br />
+[[Arctan (multivalued]] <br />
+[[Arccsc (multivalued)]] <br />
+[[Arcsec (multivalued)]] <br />
+[[Arccot (multivalued)]] <br />
+{{:Trigonometric functions footer}}
+{{:Inverse trigonometric functions footer}}
+{{:Hyperbolic trigonometric functions footer}}
+{{:Inverse hyperbolic trigonometric functions footer}}
+{{:Depreciated trigonometric functions footer}}
+===Depreciated trigonometric functions===
+[[Arccosvercosine]]<br />
+[[Arccoversine]]<br />
+[[Archavercosine]]<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.
+<div class="grid">
+<div class="grid"><center>[[File:hurwitzzetathumb.png|45px|link=Hurwitz zeta]]<br />
+[[Hurwitz zeta]]</center></div>
+<div class="grid"><center>[[File:jacobizetathumb.png|45px|link=Jacobi zeta]]<br />
+[[Jacobi zeta]]</center></div>
+<div class="grid"><center>[[File:primezetapthumb.png|45px|link=Prime zeta P]]<br />
+[[Prime zeta P|Prime zeta $P$]]</center></div>
+<div class="grid"><center>[[File:riemannzetathumb.png|45px|link=Riemann zeta]]<br />
+[[Riemann zeta]]</center></div>
+<div class="grid"><center>[[File:weierstrasszetathumb.png|45px|link=Weierstrass zeta]]<br />
+[[Weierstrass zeta]]</center></div>
+</div>
 [[Airy zeta function]]<br />
+[[Arithmetic zeta function]]<br />
 [[Arakawa-Kaneko zeta function]]<br />
 [[Arithmetic zeta function]]<br />
 [[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 />
 [[Height zeta function]]<br />
-[[Hurwitz zeta function]]<br />
 [[Igusa zeta function]]<br />
 [[Ihara zeta function]]<br />
@@ Line 116: / Line 529: @@
 [[Multiple zeta function]]<br />
 [[p-adic zeta function | $p$-adic zeta function]]<br />
-[[Prime zeta function]]<br />
+[[q-Hurwitz zeta|$q$-Hurwitz zeta]]<br />
-[[Riemann zeta function]]<br />
+[[q-zeta|$q$-zeta]]<br />
+[[Reciprocal zeta function]]<br />
 [[Ruelle zeta function]]<br />
 [[Secant zeta function]]<br />
@@ Line 124: / Line 538: @@
 [[Shintani zeta function]]<br />
 [[Witten zeta function]]<br />
+[[Z function]]<br />
-=$q$-Calculus=
+==Arithmetic Operations==
-[[q-Bessel functions | $q$-Bessel function]]<br />
+[[Addition]] <br />
-[[q-Beta function | $q$-Beta function]]<br />
+[[Division]] <br />
-[[q-Binomial function | $q$-Binomial function]]<br />
+[[Exponentiation]]<br />
-[[q-Factorial | $q$-Factorial]]<br />
+[[Hyperoperation]]<br />
-[[q-Gamma function | $q$-Gamma function]]<br />
+[[Multiplication]] <br />
-[[q-Hermite polynomial | $q$-Hermite polynomial]]<br />
+[[Subtraction]] <br />
-[[q-Pochhammer symbol | $q$-Pochhammer symbol]]<br />
+[[Tetration]]<br />
 =Special constants=
-[[Apéry's constant]]<br />
+[[Algebraic]] <br />
-[[Bernoulli numbers]]<br />
+[[Transcendental]]
-[[Chaitin's constant]]<br />
+{| class="wikitable sortable"
-[[Conway's constant]]<br />
+|-
-[[E | $e$]]<br />
+! Name !! Notation !! Value
-[[Euler-Mascheroni constant]] <br />
+|-
-[[Feigenbaum constants]]<br />
+| [[Apéry's constant]] || $\zeta(3)$  || $1.2020569031595942854 \ldots$
-[[Fransén–Robinson constant]]<br />
+|-
-[[Golden ratio]] <br />
+| [[Artin constant]] || $C_{\mathrm{Artin}}$ || $0.3739558136\ldots$
-[[Khinchin's constant]]<br />
+|-
-[[Lévy's constant]]<br />
+| [[Brun's constant]] || $B_2$ || $1.9021605831\ldots$
-[[Mills' constant]]<br />
+|-
-[[Pi | $\pi$]]<br />
+| [[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]] || || $0.23571113171923...$
+|-
+| [[E | Euler's number]] || $e$ || $2.71828182846 \ldots$
+|-
+| [[Erdős-Borwein Constant]] || $E$ || $1.606695152415291763\ldots$
+|-
+| [[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=
+[[Algebraic number]] $\overline{\mathbb{Q}}$<br />
+[[Complex number]] $\mathbb{C}$<br />
+[[Integer]] $\mathbb{Z}$<br />
+[[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=
+[[Absolute maximum]]<br />
+[[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 />
-[[Ratio_test | Ratio test]]
+[[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 />
+[[Weierstrass factorization]]
 =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 />
@@ Line 160: / Line 742: @@
 [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://people.math.sfu.ca/~cbm/aands/ 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://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]<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

External Links

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