NumExp: Numerical epsilon expansion of hypergeometric functions

TL;DR

The paper introduces NumExp, a numerical package that efficiently expands hypergeometric functions in a small parameter, facilitating calculations in dimensional regularization for loop integrals.

Contribution

NumExp provides a versatile, numerical method for expanding hypergeometric functions as Laurent series, applicable to a wide range of parameters and useful in regularization contexts.

Findings

Successfully evaluates hypergeometric functions numerically.

Extracts divergent and finite parts simultaneously.

Works for a broad class 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.