List of Special Functions

Introduction

This article lists the special functions the corpus introduces. A special function is a named solution of a differential equation or of a functional equation that the elementary functions do not solve, and a row below names one, records the equation it satisfies and the family it belongs to, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The functions are grouped by the layer that introduces them: the elementary transcendentals, which are the degenerate cases against which the others are measured; the gamma and beta family and the error functions; the zeta and number-theoretic functions; the orthogonal polynomials and spherical functions; the Bessel, Airy and hypergeometric functions; and the elliptic integrals and elliptic functions. The rows that record a boundary — the Lambert $W$, which satisfies no linear equation; the gamma function, which has no zeros; the elliptic functions, which are not single-valued without their lattice — stand beside them as non-examples.

The Elementary Transcendental Functions

Object	The equation it solves	Introduced in
the exponential $e^x$	$y' = y$, $y(0) = 1$; the sum $\sum x^n/n!$	Real Special Functions
the logarithm $\ln x$	$y' = 1/x$, $y(1) = 0$; the inverse of the exponential	Real Special Functions
the trigonometric functions $\sin$, $\cos$	$y'' + y = 0$; defined by their power series and the differential system	Real Special Functions
the hyperbolic functions $\sinh$, $\cosh$, $\tanh$	$y'' - y = 0$; the exponential in its even and odd parts	Real Special Functions
the inverse trigonometric and inverse hyperbolic functions	the equations $y' = 1/\sqrt{1-x^2}$, $y' = 1/\sqrt{1+x^2}$	Real Special Functions
general powers $a^x$	$y' = (\ln a) y$; the composition of the exponential and the logarithm	Real Special Functions
the complex exponential, logarithm and trigonometric functions	the same equations, with the multi-valuedness of the logarithm on $\mathbb C^\times$	Complex Special Functions
the Lambert $W$	$W e^W = x$; a functional equation, not a linear differential equation	Real Special Functions; Complex Special Functions

The Gamma and Beta Family

Object	The equation or the functional equation it satisfies	Introduced in
the gamma function $\Gamma$	$\Gamma(z+1) = z\Gamma(z)$, $\Gamma(1) = 1$; the integral $\int_0^\infty t^{z-1}e^{-t}dt$	Real Special Functions; Complex Special Functions
the beta function $B$	$B(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y)$; the integral over the simplex	Real Special Functions
the digamma function $\psi$	$\psi = \Gamma'/\Gamma$, the logarithmic derivative of the gamma function	Dual-Numbers Special Functions
the error function $\operatorname{erf}$	$\operatorname{erf}' = 2e^{-x^2}/\sqrt\pi$; the incomplete gamma of order $1/2$	Real Special Functions; Complex Special Functions
the complementary and imaginary error functions	the complements and the analytic continuation of $\operatorname{erf}$	Real Special Functions
the Dawson function	$F' + 2xF = 1$; the imaginary error function	Real Special Functions
the Fresnel integrals	$C' = \cos(\pi x^2/2)$, $S' = \sin(\pi x^2/2)$	Real Special Functions
the incomplete gamma and beta functions	the integral representations $\gamma(s,x)$, $B(x;a,b)$, and their functional equations	Real Special Functions; Complex Special Functions
the regularised incomplete gamma and beta functions	the ratios $P$, $Q$ and $I_x(a,b)$, with values in $[0,1]$	Real Special Functions

The Zeta and Number-Theoretic Functions

Object	The equation or the product it satisfies	Introduced in
the Riemann zeta function $\zeta$	the functional equation relating $\zeta(s)$ and $\zeta(1-s)$; the Euler product over the primes	Zeta Functions; Real Special Functions
the Dirichlet eta function $\eta$	$\eta(s) = (1-2^{1-s})\zeta(s)$; the alternating series	Real Special Functions
a Dirichlet $L$-function	the functional equation of the character, and the Euler product	Zeta Functions; Analytic Number Theory
the Bernoulli numbers $B_n$	the generating function $t/(e^t-1) = \sum B_n t^n/n!$	Real Special Functions
the Bernoulli polynomials	$B_n(x+1) - B_n(x) = nx^{n-1}$; the generating function	Real Special Functions
the Euler numbers $E_n$	$\operatorname{sech} t = \sum E_n t^n/n!$; the tangent numbers	Real Special Functions
the Möbius and Euler totient functions	the Dirichlet convolution $\mu * 1 = \varepsilon$, and the Dirichlet series	Combinatorial Functions and Generating Functions

Orthogonal Polynomials and Spherical Functions

Object	The differential equation it solves	Introduced in
the Legendre polynomials $P_n$	$(1-x^2)y'' - 2xy' + n(n+1)y = 0$; the Legendre equation	Real Special Functions; Ordinary Differential Equations
the Chebyshev polynomials $T_n$	$(1-x^2)y'' - xy' + n^2y = 0$; the Chebyshev equation	Real Special Functions; Complex Special Functions
the Hermite polynomials $H_n$	$y'' - 2xy' + 2ny = 0$; the Hermite equation, and the Fourier transform's eigenfunctions	Real Special Functions; Fourier Analysis on Euclidean Spaces
the Laguerre polynomials $L_n$	$xy'' + (1-x)y' + ny = 0$; the Laguerre equation	Real Special Functions; Complex Special Functions
the Gegenbauer polynomials $C_n^{(\lambda)}$	the Gegenbauer equation; the Chebyshev and Legendre cases at $\lambda = 1$ and $\lambda = 1/2$	Symmetric Tensors and Spherical Harmonics
the associated Legendre functions $P_n^m$	the associated Legendre equation, with the $(1-x^2)^{-m/2}$ factor	Symmetric Tensors and Spherical Harmonics
the spherical harmonics $Y_l^m$	the Laplace equation on the sphere; the eigenfunctions of the spherical Laplacian	Symmetric Tensors and Spherical Harmonics

Bessel, Airy and Hypergeometric Functions

Object	The equation it solves	Introduced in
the Bessel function $J_\nu$	$x^2y'' + xy' + (x^2-\nu^2)y = 0$; Bessel's equation	Real Special Functions; Ordinary Differential Equations
the Bessel function of the second kind $Y_\nu$	the same equation, the second solution at an integer order	Real Special Functions
the modified Bessel functions $I_\nu$, $K_\nu$	$x^2y'' + xy' - (x^2+\nu^2)y = 0$	Real Special Functions
the Airy function $\operatorname{Ai}$	$y'' - xy = 0$; the Airy equation, solved by the Fourier integral	Real Special Functions; Complex Special Functions
the hypergeometric function ${}_2F_1$	the hypergeometric equation $x(1-x)y'' + (c-(a+b+1)x)y' - aby = 0$	Real Special Functions; Complex Special Functions
the confluent hypergeometric and Whittaker functions	the confluent equation obtained as $c \to \infty$	Complex Special Functions
the generalised hypergeometric function ${}_pF_q$	the generalised equation of order $p$, $q$	Real Special Functions
the complete elliptic integrals $K$, $E$	the hypergeometric cases ${}_2F_1(1/2,1/2;1;m)$ and ${}_2F_1(-1/2,1/2;1;m)$	Real Special Functions; Complex Special Functions

Elliptic Integrals and Elliptic Functions

Object	The equation or the property it satisfies	Introduced in
the incomplete elliptic integrals $F$, $E$, $\Pi$	the integral forms, and the addition theorems of the elliptic theory	Real Special Functions; Complex Special Functions
the Jacobi elliptic functions $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$	the addition theorems and the double periodicity in the lattice	Elliptic Functions and Integrals (planned)
the Weierstrass function $\wp$	$\wp'^2 = 4\wp^3 - g_2\wp - g_3$; the associated elliptic curve	Elliptic Functions and Integrals (planned)
the elliptic modular function $j$	the modular invariance $j(\gamma\tau) = j(\tau)$ and the $j$-invariant of the curve	Elliptic Functions and Integrals (planned)

Boundary Cases

Object	The boundary it records	Introduced in
the gamma function at its poles	has simple poles at $0,-1,-2,\dots$ and no zeros; the reciprocal $1/\Gamma$ is entire	Complex Special Functions
the elementary functions as hypergeometric cases	the exponential, the logarithm and the trigonometric functions are degenerate ${}_2F_1$ cases, which is why they carry no independent equation	Real Special Functions
the Lambert $W$	satisfies a functional equation, not a linear differential equation, and is multiply valued	Real Special Functions
the Bessel functions at an integer order	the second solution $Y_n$ requires the limit of $Y_\nu$, the logarithmic case	Real Special Functions
the elliptic functions	not single-valued on $\mathbb C$; defined on the torus $\mathbb C/\Lambda$ by their lattice	Elliptic Functions and Integrals (planned)

Summary

This list gathers the special functions of the corpus: the elementary transcendentals and the Lambert $W$; the gamma, beta, error, incomplete and Fresnel functions; the zeta function with its Dirichlet relatives and the Bernoulli and Euler numbers; the Legendre, Chebyshev, Hermite, Laguerre, Gegenbauer and spherical functions; the Bessel, Airy and hypergeometric functions; and the elliptic integrals, with the elliptic functions that the planned article introduces. Each row records the equation the function solves, and the closing table records the boundaries of the class.

Summary of Notation

The objects are named by their standard letters; the few symbols used in the tables are collected here.

Symbol	Meaning
$\Gamma$, $B$	the gamma and beta functions
$\zeta$, $\eta$, $L(s,\chi)$	the Riemann zeta, Dirichlet eta and Dirichlet $L$-functions
$P_n, T_n, H_n, L_n$	Legendre, Chebyshev, Hermite, Laguerre polynomials
$J_\nu$, $I_\nu$, $\operatorname{Ai}$	Bessel, modified Bessel and Airy functions
${}_2F_1$, ${}_pF_q$	hypergeometric and generalised hypergeometric functions
$K$, $E$, $\Pi$	complete and incomplete elliptic integrals
$W$	the Lambert $W$ function