NumExp: Numerical epsilon expansion of hypergeometric functions

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.

💡 Research Summary

The paper introduces NumExp, a software package that performs numerical ε‑expansions of hypergeometric and related transcendental functions, a task that is central to dimensional regularization in multi‑loop quantum field‑theory calculations. Traditional approaches rely on symbolic manipulation (e.g., Mathematica, FORM, HyperInt) to obtain Laurent series in the regularization parameter ε. Those methods often encounter severe limitations: convergence problems when parameters lie near singularities, combinatorial explosion of intermediate expressions, and a lack of flexibility for multivariate hypergeometric functions. NumExp circumvents these issues by abandoning symbolic series generation and instead evaluating the coefficients of the Laurent expansion directly through a high‑precision finite‑difference scheme.

Core Methodology

1. Laurent Representation – Any well‑regularized hypergeometric function (F(ε)) is expressed as
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