Software Index

Software Cross Index

	Open Source													With Book						Commercial
Package																									See Also
Language	C	Int.	C	Ftn	C	C++	C	Int.	C$\mathrm{\#}$	Int. C	Ftn	Int.	Py	C	C Java	C C++ Ftn	C Ftn Mma	Ftn	Ftn	C Ftn Java	Int.	Int.	Int.	C Ftn
4 Elementary Functions
4.48(ii) Interval Arithmetic	✓					✓						✓	✓								✓	✓	a		CoStLy
4.48(iii) General Precision	✓					✓	✓	✓				✓	✓								✓	✓	a		REDUCE
4.48(iv) Lambert $W$-Function	✓					✓			✓			✓	✓								✓	✓	a
4.48(v) Testing
5 Gamma Function
5.24(ii) $\mathrm{\Gamma }\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM
5.24(iii) $\psi \left(x\right)$, ${\psi }^{(n)}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓			✓		✓	✓	✓	✓	✓	✓
5.24(iv) $\mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, ${\psi }^{(n)}\left(z\right)$, $z\in ℂ$	✓	✓		✓		✓			✓	✓	✓	✓	✓	✓				✓	✓	✓	✓	✓	✓	✓
5.24(v) $\mathrm{B}(a,b)$, $a,b\in ℝ$	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓	✓			✓		✓	✓	✓	✓	✓
5.24(vi) $\mathrm{B}(a,b)$, $a,b\in ℂ$	✓	✓		✓							✓	a	✓							✓	✓	✓	a
6 Exponential, Logarithmic, Sine, and Cosine Integrals
6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓		✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$	✓	✓				✓			✓			✓	✓	✓							✓	✓	✓
7 Error Functions, Dawson’s and Fresnel Integrals
7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$	✓								✓			a	✓	✓							✓	✓	✓
7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$	✓		✓						✓			a	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$	✓											a	✓								✓	✓	✓
7.25(vi) $ℱ\left(x\right)$, $G\left(x\right)$, $𝖴(x,t)$, $𝖵(x,t)$, $x\in ℝ$			✓	✓	✓						✓			✓	✓	✓	✓			✓	✓	✓	✓
7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$																					✓	✓	✓
8 Incomplete Gamma and Related Functions
8.28(ii) $\gamma (a,x)$, $\mathrm{\Gamma }(a,x)$, ${\gamma }^{\ast }(a,x)$, $P(a,x)$, $Q(a,x)$, $x,a\in ℝ$	✓		✓	✓	✓	✓			✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
8.28(iii) $\gamma (a,x)$, $\mathrm{\Gamma }(a,x)$, ${\gamma }^{\ast }(a,x)$, $P(a,x)$, $Q(a,x)$, $x,a\in ℂ$	✓		✓	✓	✓						✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓
8.28(iv) ${\mathrm{B}}_x(a,b)$, $I_x(a,b)$, $x,a,b\in ℝ$	✓								✓
8.28(v) ${\mathrm{B}}_z(a,b)$, $I_z(a,b)$, $z,a,b\in ℂ$	✓											a	✓									✓
8.28(vi) $E_p\left(x\right)$, $x\in ℝ$, $p\in \mathbb{Z}$	✓		✓		✓				✓		✓	✓	✓	✓	✓	✓			✓	✓	✓	✓	✓
8.28(vii) $E_p\left(z\right)$, $z,p\in ℂ$	✓											✓	✓								✓	✓	✓
9 Airy and Related Functions
9.20(ii) $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓
9.20(iii) $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$, $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$, $z\in ℂ$	✓	✓						✓			✓	a	✓	✓							✓	✓	✓	✓
9.20(iv) Zeros of …	✓				✓							a	✓								✓	✓	a
9.20(v) Integrals of …	✓											a	✓	✓					✓		✓	✓	a
9.20(vi) Scorer Functions	✓											a	✓
10 Bessel Functions
10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)	✓	✓	✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM, NMS
10.77(iii) Bessel Functions–Real Order and Argument	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	NMS
10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions	✓	✓	✓									a	✓	✓			✓	✓	✓	✓	✓	✓	✓
10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)	✓	✓									✓	a	✓						✓	✓	✓	✓	✓
10.77(vi) Bessel Functions–Imaginary Order and Real Argument	✓	✓										✓	✓								✓	✓	✓
10.77(viii) Bessel Functions–Complex Order and Argument	✓	✓										✓	✓								✓	✓	✓
10.77(ix) Integrals of Bessel Functions	✓										✓	a	✓	✓					✓		✓	✓	a
10.77(x) Zeros of Bessel Functions					✓							a	✓	✓	✓				✓		✓	✓	a
11 Struve and Related Functions
11.16(ii) ${𝐇}_{\nu }\left(z\right)$, ${𝐊}_{\nu }\left(z\right)$			✓									a	✓	✓			✓		✓		✓	✓	a
11.16(iii) Integrals of …														✓					✓		✓	✓	a
11.16(iv) $s_{\mu ,\nu }\left(z\right)$, $S_{\mu ,\nu }\left(z\right)$												a	✓				✓				✓		a
11.16(v) ${𝐉}_{\nu }\left(z\right)$, ${𝐄}_{\nu }\left(z\right)$, ${𝐀}_{\nu }\left(z\right)$												a	✓				✓				✓	✓	a
11.16(vi) Integrals of …														✓							✓	✓	a
12 Parabolic Cylinder Functions
12.21(ii) $U(a,x)$, $V(a,x)$, $\overline{U}(a,x)$, $W(a,x)$, $x,a\in ℝ$						✓						a	✓				✓		✓		✓	✓	a
12.21(iii) $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, $W(a,z)$, $z,a\in ℂ$												a	✓								✓	✓	a
13 Confluent Hypergeometric Functions
13.32(ii) $M(a,b,x)$, $U(a,b,x)$, $𝐌(a,b,x)$, $M_{\kappa ,\mu }\left(x\right)$, $W_{\kappa ,\mu }\left(x\right)$, $x,a,b\in ℝ$	✓		✓	✓	✓	✓						✓	✓	✓			✓		✓		✓	✓	a
13.32(iii) $M(a,b,z)$, $U(a,b,z)$, $𝐌(a,b,z)$, $M_{\kappa ,\mu }\left(z\right)$, $W_{\kappa ,\mu }\left(z\right)$, $z,a,b\in ℂ$	✓											✓	✓	✓					✓		✓	✓	a
14 Legendre and Related Functions
14.34(ii) ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(x\right)$, $Q_{\nu }\left(x\right)$, $x,\nu \in ℝ$	✓				✓	✓					✓	a	✓	✓		✓	✓		✓		✓	✓	✓	✓
14.34(iii) ${𝖯}_{\nu }\left(z\right)$, ${𝖰}_{\nu }\left(z\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$, $z,\nu \in ℂ$	✓											a	✓	✓							✓	✓	a
14.34(iv) ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$					✓									✓			✓
15 Hypergeometric Function
15.20(ii) ${}_2{}^{}F_1^{}(a,b;c;x)$, $x,a,b,c\in ℝ$	✓		✓		✓	✓						a	✓	✓			✓		✓		✓	✓	a
15.20(iii) ${}_2{}^{}F_1^{}(a,b;c;z)$, $z,a,b,c\in ℂ$	✓											a	✓			✓					✓	✓	a
16 Generalized Hypergeometric Functions & Meijer G-Function
16.27(ii) Real Arguments	✓		✓		✓	✓						a	✓								✓	✓	✓
16.27(iii) Complex Arguments	✓											a	✓								✓	✓	✓
18 Orthogonal Polynomials
18.42 Software	✓	✓				✓			✓			✓	✓								✓	✓	a		Koornwinder, Stembridge
19 Elliptic Integrals
19.39(ii) $K\left(k\right)$, $E\left(k\right)$, $0\le k^2\le 1$	✓		✓		✓	✓		✓	✓			✓	✓							✓	✓	✓	✓
19.39(iii) $F(\varphi ,k)$, $E(\varphi ,k)$, $\varphi ,k\in ℝ$	✓		✓		✓	✓		✓	✓			✓	✓			✓					✓	✓	a	✓
19.39(iv) $R_C(x,y)$, $R_F(x,y,z)$, $R_D(x,y,z)$, $R_J(x,y,z,p)$	✓				✓				✓		✓	a	✓			✓				✓				✓	Derive
20 Theta Functions
20.16(ii) Real arguments	✓											a	✓	✓			✓				✓	✓	a	✓
20.16(iii) Complex arguments	✓											a	✓	✓							✓	✓	a
21 Multidimensional Theta Functions
21.11 Software																					✓	✓	a		JTEM
22 Jacobian Elliptic Functions
22.22(ii) Real Argument	✓		✓		✓	✓		✓				✓	✓	✓		✓	✓		✓	✓	✓	✓	✓	✓
22.22(iii) Complex Argument	✓					✓						✓	✓	✓						✓	✓	✓	a	✓
23 Weierstrass Elliptic and Modular Functions
23.24(ii) Real Argument																					✓	✓	a
23.24(iii) Complex Argument														✓						✓	✓	✓	a
24 Bernoulli and Euler Polynomials
24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
25 Zeta and Related Functions
25.21(ii) $\zeta \left(s\right)$, $s\in ℝ$	✓		✓		✓	✓	✓	✓	✓			✓	✓	✓			✓				✓	✓	✓
25.21(iii) $\zeta \left(s\right)$, $s\in ℂ$	✓					✓			✓			✓	✓	✓							✓	✓	✓
25.21(iv) $\zeta (s,a)$	✓				✓	✓						✓	✓	✓			✓				✓	✓	a
25.21(v) ${\mathrm{Li}}_2\left(z\right)$, ${\mathrm{Li}}_s\left(z\right)$	✓		✓		✓	✓	✓		✓			✓	✓	✓			✓				✓	✓	✓
25.21(vi) Clausen’s Integral					✓				✓			a	✓									✓			NetNUMPAC
25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
25.21(viii) Lerch’s Transcendent												a	✓								✓	✓	a
25.21(ix) Dirichlet $L$-series	✓											a	✓									✓
26 Combinatorial Analysis
26.22 Software						✓						✓									✓	✓	a		Wolfram’s Mathworld, Zeilberger
27 Functions of Number Theory
27.22 Software	✓					✓				✓		✓									✓	✓	a
28 Mathieu Functions and Hill’s Equation
28.36(ii) Exponents, Eigenvalues														✓					✓	✓	✓	✓	a
28.36(iii) Mathieu Functions														✓					✓	✓	✓	✓	a		Van Buren
30 Spheroidal Wave Functions
30.18(ii) Eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$														✓			✓		✓			✓
30.18(iii) Wave Functions														✓			✓		✓			✓			Van Buren
33 Coulomb Functions
33.26(ii) Real arguments	✓				✓				✓			a	✓	✓			✓				✓		a
33.26(iii) Complex arguments	✓											a	✓
34 3j, 6j, 9j Symbols
34.15 Software					✓	✓			✓		✓	✓		✓			✓				✓	✓	a
35 Functions of Matrix Argument
35.12 Software																									Zeilberger

‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support.

A Classification of Software

In the list below we identify four main sources of software for computing special functions. Please see our Software Indexing Policy for rules that govern the indexing of software in the DLMF.

Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

Software Associated with Books.

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

Software Repositories

The following are web-based software repositories with significant holdings in the area of special functions. Many research software packages are found here, as well as some open source software collections.

Collected Algorithms of the ACM

Software published by the journal ACM Transactions on Mathematical Software (TOMS).

Computer Physics Communications Program Library

Software associated with papers published in the journal Computer Physics Communications.

netlib

A collection of mathematical software, papers, and databases produced by the numerical analysis research community.

Guide to Available Mathematical Software

A cross index of mathematical software in use at NIST.