NumExp: Numerical epsilon expansion of hypergeometric functions

Abstract

It is demonstrated that the well-regularized hypergeometric functions can be evaluated directly and numerically. The package NumExp is presented for expanding hypergeometric functions and/or other transcendental functions in a small regularization parameter. The hypergeometric function is expressed as a Laurent series in the regularization parameter and the coefficients are evaluated numerically by using the multi-precision finite difference method. This elaborate expansion method works for a wide variety of hypergeometric functions, which are needed in the context of dimensional regularization for loop integrals. The divergent and finite parts can be extracted from the final result easily and simultaneously. In addition, there is almost no restriction on the parameters of hypergeometric functions. Program summaryProgram title:NumExpCatalogue identifier: AEPE_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPE_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1682No. of bytes in distributed program, including test data, etc.: 18431Distribution format: tar.gzProgramming language: Mathematica and/or Python.Computer: Any computer where Mathematica or Python is running.Operating system: Linux, Windows.Classification: 4.4, 5, 11.1.External routines: mpmath library (for Python)Nature of problem:Expansion of hypergeometric functions and/or other transcendental functions in a small parameter ε. These expansions are needed in the context of dimensional regularization for loop integrals.Solution method:The hypergeometric function is expressed as a Laurent series in the regularization parameter ε, where the coefficients are evaluated numerically by the multi-precision finite difference method.Restrictions:The calculation may be inefficient if the arguments of hypergeometric functions are close to the convergent boundaries.Running time:Generally it is less than a few seconds, depending on the complexity of the problem.