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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities


Author: Avram Sidi
Journal: Math. Comp. 81 (2012), 2159-2173
MSC (2010): Primary 30E15, 40A25, 41A60, 65B15, 65D30
Published electronically: April 10, 2012
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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

  $\displaystyle f(x)\sim K\,(x-a)^{-1}+\sum ^{\infty }_{s=0}c_s(x-a)^{\gamma _s}$$\displaystyle \quad \text {as}\ x\to a+,$    
  $\displaystyle f(x)\sim L\,(b-x)^{-1}+\sum ^{\infty }_{s=0}d_s(b-x)^{\delta _s}$$\displaystyle \quad \text {as}\ x\to b-,$    

where $ K,L$, and $ c_s, d_s$, $ s=0,1,\ldots ,$ are some constants, $ \vert K\vert+\vert L\vert\neq 0,$ and $ \gamma _s$ and $ \delta _s$ are distinct, arbitrary and, in general, complex, and different from $ -1$, and satisfy

$\displaystyle \Re \gamma _0\leq \Re \gamma _1\leq \cdots , \ \ \lim _{s\to \inf... ...\leq \Re \delta _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \delta _s=+\infty .$

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

$\displaystyle h\sum ^{n-1}_{i=1}f(a+ih)\sim I[f]$ $\displaystyle +K\,(C -\log h) + \sum ^{\infty }_{\substack {s=0\\ \gamma _s\not \in \{2,4,\ldots \}}}c_s \zeta (-\gamma _s)\,h^{\gamma _s+1}$    
  $\displaystyle +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}$$\displaystyle \quad \text {as $h\to 0$},$    

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.


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Additional Information

Avram Sidi
Affiliation: Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel
Email: asidi@cs.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02597-X
PII: S 0025-5718(2012)02597-X
Keywords: Euler–Maclaurin expansions, trapezoidal rule, endpoint singularities, Hadamard finite part, asymptotic expansions.
Received by editor(s): November 16, 2010
Received by editor(s) in revised form: April 17, 2011
Published electronically: April 10, 2012
Additional Notes: This research was supported in part by the United States–Israel Binational Science Foundation grant no. 2008399.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.