Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities

Abstract:

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the divergent finite-range integrals $\int ^b_af(x) dx$, where $f\in C^{\infty }(a,b)$ but can have arbitrary algebraic singularities at one or both endpoints. We assume that $f(x)$ has asymptotic expansions of the general forms \begin{align*} &f(x)\sim K (x-a)^{-1}+\sum ^{\infty }_{s=0}c_s(x-a)^{\gamma _s} \quad \text {as}\ x\to a+,\\ &f(x)\sim L (b-x)^{-1}+\sum ^{\infty }_{s=0}d_s(b-x)^{\delta _s} \quad \text {as}\ x\to b-, \end{align*} where $K,L$, and $c_s, d_s$, $s=0,1,\ldots ,$ are some constants, $|K|+|L|\neq 0,$ and $\gamma _s$ and $\delta _s$ are distinct, arbitrary and, in general, complex, and different from $-1$, and satisfy \[ \Re \gamma _0\leq \Re \gamma _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \gamma _s=+\infty ;\quad \Re \delta _0\leq \Re \delta _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \delta _s=+\infty .\] Hence the integral $\int ^b_af(x) dx$ exists in the sense of Hadamard finite part. The results we obtain in this work extend some of the results in [A. Sidi, Numer. Math. 98 (2004), pp. 371–387] that pertain to the cases in which $K=L=0.$ They are expressed in very simple terms based only on the asymptotic expansions of $f(x)$ as $x\to a+$ and $x\to b-$. With $h=(b-a)/n$, where $n$ is a positive integer, one of these results reads \begin{align*} h\sum ^{n-1}_{i=1}f(a+ih)\sim I[f]&+K (C -\log h) + \sum ^{\infty }_{\substack {s=0\\ \gamma _s\not \in \{2,4,\ldots \}}}c_s \zeta (-\gamma _s) h^{\gamma _s+1}\\ &+L (C -\log h) +\sum ^{\infty }_{\substack {s=0\\ \delta _s\not \in \{2,4,\ldots \}}}d_s\zeta (-\delta _s) h^{\delta _s+1}\quad \text {as $h\to 0$}, \end{align*} where $I[f]$ is the Hadamard finite part of $\int ^b_af(x) dx$, $C$ is Euler’s constant and $\zeta (z)$ is the Riemann Zeta function. We illustrate the results with an example.