Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions

Abstract

We consider the numerical computation of finite-range singular integrals $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$ and $f(x)\in C^\infty(\mathbb{R}_t)$ is $T$-periodic with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$, $T=b-a$. Using a generalization of the Euler--Maclaurin expansion developed in [A. Sidi, {Euler--Maclaurin} expansions for integrals with arbitrary algebraic endpoint singularities. {\em Math. Comp.}, 81:2159--2173, 2012], we unify the treatment of these integrals. For each $m$, we develop a number of numerical quadrature formulas $\widehat{T}^{(s)}_{m,n}[f]$ of trapezoidal type for $I[f]$. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case $m=3$, and these are \begin{align*} \widehat{T}^{(0)}_{3,n}[f]&=h\sum^{n-1}_{j=1}f(t+jh)-\frac{\pi^2}{3}\,g'(t)\,h^{-1} +\frac{1}{6}\,g'''(t)\,h, \quad h=\frac{T}{n}, \widehat{T}^{(1)}_{3,n}[f]&=h\sum^n_{j=1}f(t+jh-h/2)-\pi^2\,g'(t)\,h^{-1},\quad h=\frac{T}{n}, \widehat{T}^{(2)}_{3,n}[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4),\quad h=\frac{T}{n}.\end{align*}