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.

fixed minor bugs; to appear in Computer Physics Communications

References in corpus (20)

• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• SecDec: A general program for sector decomposition
• On the Generalized Harmonic Polylogarithms of One Complex Variable
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• $\{Q\bar{q}\}\{{\bar{Q}}^{{(}^{\prime })}q\}$ molecular states
• Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
• HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables
• Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Poschl-Teller-Ginocchio potential wave functions
• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
• Finding new relationships between hypergeometric functions by evaluating Feynman integrals
• Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions
• Hypergeometric representation of the two-loop equal mass sunrise diagram
• On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters
• Optimized Negative Dimensional Integration Method (NDIM) and multiloop Feynman diagram calculation
• A massive Feynman integral and some reduction relations for Appell functions
• Acceleration of generalized hypergeometric functions through precise remainder asymptotics
• Definite integrals by the method of brackets. Part 1
• A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams
• Analytic calculation of doubly heavy hadron spectral density in coordinate space
• Partial sums of hypergeometric series of unit argument

Cited by in corpus (15)

• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
• HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables
• 3d Abelian Gauge Theories at the Boundary
• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: $F_D$ and $F_S$ Horn-type hypergeometric functions of three variables
• Resummed photon spectrum from dark matter annihilation for intermediate and narrow energy resolution
• A new approach to the epsilon expansion of generalized hypergeometric functions
• Towards the QED beta function and renormalons at $1/N_f^2$ and $1/N_f^3$
• Analytic Coupling Structure of Large $N_f$ (Super) QED and QCD
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• q-derivatives of multivariable q-hypergeometric function with respect to their parameters
• Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d, II: Special kinematics
• Hypergeometric presentation for one-loop contributing to $H\to Z\gamma$