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.

Registration 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.

@@ Line 1: / Line 1: @@
 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@gmail.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=
@@ Line 10: / Line 10: @@
 <div class="grid"><center>[[File:dedekindetathumb.png|45px|link=Dedekind eta]]<br />
 [[Dedekind eta]]</center></div>
-<div class="grid"><center>[[File:devilstaircasethumb.png|45px|link=Devil's staircase]]<br />
-[[Devil's staircase]]</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>[[File:expthumb.png|45px|link=Exponential]]<br />
+<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 />
@@ Line 28: / Line 26: @@
 <div class="grid"><center>[[File:lambertwthumb.png|45px|link=Lambert W]]<br />
 [[Lambert W|Lambert $W$]]</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>[[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:riemannsiegelzthumb.png|45px|link=Riemann-Siegel Z]]<br />
-[[Riemann-Siegel Z|Riemann-Siegel $Z$]]</center></div>
 <div class="grid"><center>[[File:riemannxithumb.png|45px|link=Riemann xi]]<br />
 [[Riemann xi|Riemann $\xi$]]</center></div>
@@ Line 44: / Line 38: @@
 </div>
-[[Arithmetic functions]]<br />
+[[Bickley-Naylor]]<br />
-[[Bessel-Clifford function]]<br />
-[[Böhmer integral]]<br />
-[[Buchstab function]]<br />
 [[Carlitz-Goss Gamma function]]<br />
-[[Clausen functions]]<br />
+[[Clausen cosine]]<br />
-[[Dawson function]]<br />
+[[Clausen sine]]<br />
 [[Debye function]]<br />
-[[Determinant]]<br />
-[[Dickman–de Bruijn function]]<br />
 [[Dirichlet function]]<br />
 [[Discriminant]]<br />
 [[Distance to integers]]<br />
 [[Eisenstein series]]<br />
-[[Entire]]<br />
-[[Faddeeva function]]<br />
 [[Heaviside step function]]<br />
 [[Hermite function]]<br />
-[[Jacobi theta 1]]<br />
-[[Jacobi theta 2]]<br />
-[[Jacobi theta 3]]<br />
-[[Jacobi theta 4]]<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 />
-[[Modified Struve function]]<br />
+[[Mittag-Leffler]]<br />
 [[Modular form]]<br />
 [[Mock modular forms]]<br />
 [[Mock theta functions]]<br />
 [[Nome]]<br />
-[[p-adic L function|$p$-adic $L$ function]]<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 />
 [[Singular function]]<br />
-[[Struve function]]<br />
 [[Theta functions]]<br />
+[[Unit step function]]<br />
 [[Voight function]]<br />
 [[Volterra function]]<br />
 [[Weierstrass elementary factors]]<br />
-[[Weierstrass function]]<br />
 ==Bessel functions and friends==
@@ Line 99: / Line 82: @@
 {{:Kelvin functions footer}}
+[[Modified Struve function]]<br />
 [[Riccati-Bessel S | Riccati-Bessel $S_n$]]<br />
 [[Riccati-Bessel C | Riccati-Bessel $C_n$]]<br />
@@ Line 104: / Line 90: @@
 [[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==
@@ Line 112: / Line 100: @@
 ==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>
@@ Line 136: / Line 128: @@
 [[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==
@@ Line 161: / Line 158: @@
 ==Error function and friends==
-[[Dawson F]]<br />
+<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}}
@@ Line 222: / Line 222: @@
 ==Hypergeometric series==
-<div class="grid">
+{{:Hypergeometric functions footer}}
-<div class="grid"><center>[[File:hypergeometricthumb.png|45px|link=Hypergeometric pFq]]<br />
-[[Hypergeometric pFq|Hypergeometric ${}_pF_q$]]</center></div>
-</div>
-[[Hypergeometric 0F0]]<br />
+[[Hypergeometric 0F3]]<br />
-[[Hypergeometric 1F0]]<br />
-[[Hypergeometric 0F1]]<br />
 [[Hypergeometric 1F1]]<br />
-[[Hypergeometric 2F1]]<br />
 [[Hypergeometric 1F2]]<br />
 [[Hypergeometric 2F0]]<br />
-[[Hypergeometric 2F1]]<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==
-<div class="grid">
+{{:Logarithm and friends footer}}
-<div class="grid"><center>[[File:Dilogarithmthumb.png|45px|link=Dilogarithm]]<br />
-[[Dilogarithm]]</center></div>
-<div class="grid2"><center>[[File:lithumb.png|45px|link=Logarithmic integral]]<br />
-[[Logarithmic integral|Log integral]]</center></div>
-<div class="grid"><center>[[File:logthumb.png|45px|link=Logarithm]]<br />
-[[Logarithm]]</center></div>
-<div class="grid"><center>[[File:polylog1thumb.png|45px|link=Polylogarithm]]<br />
-[[Polylogarithm]]</center></div>
-</div>
+[[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==
@@ Line 262: / Line 279: @@
 [[Mellin transform]]<br />
 [[Radon transform]]<br />
+[[Two-dimensional Laplace transform]]<br />
 ==Polynomials==
@@ Line 268: / Line 286: @@
 [[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 />
@@ Line 278: / Line 341: @@
 [[Cyclotomic polynomials]]<br />
 [[Denisyuk polynomials]]<br />
+[[Eberlein polynomials]]<br />
 [[Hahn polynomial]]<br />
 [[Humbert polynomials]]<br />
@@ Line 312: / Line 376: @@
 ===$q$-special functions===
-[[Basic hypergeometric series phi|Basic hypergeometric series $\phi$]]<br />
+[[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 />
@@ Line 338: / Line 402: @@
 [[q-Gaussian distribution | $q$-Gaussian distribution]]<br />
 [[q-Hermite polynomial | $q$-Hermite polynomial]]<br />
-[[q-Hypergeometric function | $q$-Hypergeometric function]]<br />
+[[q-Hurwitz zeta|$q$-Hurwitz zeta]]<br />
 [[q-Pochhammer | $q$-Pochhammer symbol]]<br />
 [[q-Pollaczek polynomial]]<br />
@@ Line 347: / Line 411: @@
 [[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 />
@@ Line 356: / Line 421: @@
 [[Dirichlet L-function | Dirichlet $L$-function]]<br />
+[[p-adic L function|$p$-adic $L$ function]]<br />
 [[Ramanujan's sum]]<br />
 [[Riemann prime counting]]<br />
@@ Line 371: / Line 437: @@
 [[Farey sequence]]<br />
 [[Fermat numbers]]<br />
-[[Fibonacci sequence]]<br />
+[[Fibonacci numbers]]<br />
 [[Genocchi numbers]]<br />
 [[Harmonic number]]<br />
-[[Lah numbers]]<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 />
@@ Line 387: / Line 457: @@
 [[Stirling numbers of the first kind]]<br />
 [[Stirling numbers of the second kind]]<br />
+[[Sylvester's sequence]]<br />
 [[Thue-Morse sequence]]<br />
@@ Line 410: / Line 481: @@
 [[Arcvercosine]]<br />
 [[Arcversine]]<br />
-[[Coversine]]<br />
-[[Covercosine]]<br />
-[[Excosecant]]<br />
-[[Exsecant]]<br />
 [[Hacovercosine]]<br />
 [[Hacoversine]]<br />
@@ Line 462: / 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 481: / Line 550: @@
 =Special constants=
+[[Algebraic]] <br />
+[[Transcendental]]
 {| class="wikitable sortable"
 |-
@@ Line 491: / Line 562: @@
 | [[Brun's constant]] || $B_2$ || $1.9021605831\ldots$
 |-
-| [[Cahen's constant]] ||  ||
+| [[Cahen's constant]] || $C$ || $0.64341054629\ldots$
 |-
 | [[Catalan constant]] || $G$ || $0.915965594177219015054603514932384110774 \ldots$
@@ Line 542: / Line 613: @@
 |-
 | [[Khinchin's constant]] || $K$ || $2.685452001065306445309714835481795693820382293994462953051152 \ldots$
+|-
+| [[Komornik–Loreti constant]] || $q$ || $1.787231650\ldots$
 |-
 | [[Legendre's constant]] || $B$ || $1$
@@ Line 575: / Line 648: @@
 | [[Ramanujan constant]] || $R$ || $262537412640768743.9999999999992500\ldots$
 |-
-| [[Reciprocal Fibonacci constant]] || $\psi$ ||
+| [[Reciprocal Fibonacci constant]] || $\psi$ || $3.35988566624317755\ldots$
 |-
 | [[Rutherford constant]] || $K_R$ || $0.8227\ldots$
@@ Line 596: / Line 669: @@
 |-
 | [[Thue constant]] || || $0.8590997969\ldots$
+|-
+| [[Trott's constant]] || || $0.10841015\ldots$
 |-
 | [[Twin prime constant]] || ||
@@ Line 623: / Line 698: @@
 [[Absolute minimum]]<br />
 [[Analytic continuation]]<br />
+[[Arithmetic functions]]<br />
 [[Binomial series]]<br />
 [[Cell]]<br />
@@ Line 629: / Line 705: @@
 [[Difference equation of hypergeometric type]]<br />
 [[Doubly periodic function]]<br />
+[[Entire]]<br />
 [[Fundamental pair of periods]]<br />
 [[Integration by parts]]<br />
@@ Line 659: / Line 736: @@
 =External Links=
 [http://authors.library.caltech.edu/43491/ Higher transcendental functions (Bateman project)]<br />
-[http://dualaud.net/txt/rainvillespecialfunctions.php Solution manual to Rainville's "Special Functions"]<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 Combinatorics"]<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

Navigation menu

Search