Mathematica packages that do certain types of symbolic mathematics such as indefinite integral evaluation.

Simple package for replacing combinations of constants with single constants.

Last updated March 12, 2016
symbolic

The AbstractAlgebra package is a freely available complement to the book *Exploring Abstract Algebra with Mathematica*. The package supports working with (finite) groups, rings, fields, and morphisms and functions related to each of these objects. There are a large number of built-in groups (including such standard groups as $Z_n$, $U_n$ (units of $Z_n$), $S_n$, and $D_n$, as well as direct products and quotients of these) and rings (including $Z_n$, Boolean rings and lattice rings, as well as polynomial, matrix and function extension rings). One can also create functions between groups or rings and investigate if these are morphisms.

The HolonomicFunctions package allows to deal with multivariate holonomic functions and sequences. For this purpose the package can compute annihilating ideals and execute closure properties (addition, multiplication, substitutions) for such functions. An annihilating ideal represents the set of linear differential equations, linear recurrences, q-difference equations, and mixed linear equations that a given function satisfies. Summation and integration of multivariate holonomic functions can be performed via creative telescoping. As subtasks, the following functionalities have been implemented in HolonomicFunctions: computations in Ore algebras (noncommutative polynomial arithmetic with mixed difference-differential operators), noncommutative Gröbner bases, and solving of coupled linear systems of differential or... Read more.

Expands hypergeometric JFJ-1 functions around their parameters. Detailed descriptions are available at hep-ph/0507094 and arXiv:0708.2443 .

A Mathematica-package for OPEs in vertex algebras. The package Lambda is designed for calculating λ-brackets in both vertex algebras, and in SUSY vertex algebras. This is equivalent to calculating operator product expansions in two-dimensional conformal field theory. For an introduction, see http://arxiv.org/abs/1004.5264 .

Seven different packages related to mathematical physics; a package for GR-type tensor algebra, one for Virasoro algebra, one for algebra with Grassman variables, a package for Polchinski theta-function conventions, a package specialized at inverting diagonal matrices by inverting each diagonal entry separately, and two packages that deal with approximations to the Ricci-flat metric on the algebraic Calabi-Yau manifold.

RISCErgoSum is a collection of packages created at the Research Institute for Symbolic Computation (RISC), Linz, Austria. The included packages are: Asymptotics for computing asymptotic series expansions of univariate holonomic sequences; Dependencies for computing algebraic relations of C-finite sequences and multi-sequences; Engel is an implementation of q-Engel Expansion; fastZeil, the Paule/Schorn Implementation of Gosper’s and Zeilberger’s Algorithms; GeneratingFunctions for manipulations of univariate holonomic functions and sequences; GenOmega, Guo-Niu Han’s general Algorithm for MacMahon’s Partition Analysis; Guess for guessing multivariate recurrence equations; HolonomicFunctions for dealing with multivariate holonomic functions, including closure properties, summation, and integration; MultiSum for proving hypergeometric multi-sum... Read more.

Last updated January 6, 2018
symbolic

This packages is for interfacing with the Singular computer algebra system. Singular is an open-source computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory.

A package for carrying out vector calculus calculations. Related paper: http://arxiv.org/abs/1309.2561.

xAct is a suite of free and actively maintained packages for tensor computer algebra in Mathematica. xAct implements state-of-the-art algorithms for fast manipulations of indices and has been modelled on the current geometric approach to General Relativity. It is highly programmable and configurable. Since its first public release in March 2004, xAct has been intensively tested and has solved a number of hard problems in GR.