An Extension of the Euler-Maclaurin Summation Formula to Nearly Singular Functions

TL;DR

This paper extends the Euler-Maclaurin formula to effectively handle near-singular functions, enabling highly accurate numerical integration with few nodes regardless of the singularity's strength.

Contribution

The paper introduces a novel extended Euler-Maclaurin formula that combines a singular component and a jump component for near-singular functions, improving quadrature accuracy.

Findings

Achieves machine-precision accuracy in near-singular quadrature

Maintains high accuracy regardless of singularity strength

Uses few quadrature nodes for efficient computation

Abstract

A extension of the Euler-Maclaurin (E-M) formula to near-singular functions is presented. This extension is derived based on earlier generalized E-M formulas for singular functions. The new E-M formulas consists of two components: a ``singular'' component that is a continuous extension of the earlier singular E-M formulas, and a ``jump'' component associated with the discontinuity of the integral with respect to a parameter that controls near singularity. The singular component of the new E-M formulas is an asymptotic series whose coefficients depend on the Hurwitz zeta function or the digamma function. Numerical examples of near-singular quadrature based on the extended E-M formula are presented, where accuracies of machine precision are achieved insensitive to the strength of the near singularity and with a very small number of quadrature nodes.