%0 Journal Article %1 sidi2012eulermaclaurin %A Sidi, Avram %D 2012 %J Mathematics of Computation %K analysis asymptotic integral mathematics numerical series %N 280 %P 2159-2173 %R 10.1090/S0025-5718-2012-02597-X %T Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities %U http://dx.doi.org/10.1090/S0025-5718-2012-02597-X %V 81 %X In this paper, we provide the Euler-Maclaurin expansions for (offset) trapezoidal rule approximations of the divergent finite-range integrals $ ^b_af(x)\,dx$, where $ fC^ınfty(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: ... where $ K,L$, and $ c_s, d_s$, $ s=0,1,,$ are some constants, $ K\vert+L\vert0,$ and $ _s$ and $ _s$ are distinct, arbitrary and, in general, complex, and different from $ -1$, and satisfy ... Hence the integral $ ^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 $xa+$ and $ xb-$. With $ h=(b-a)/n$, where $ n$ is a positive integer, one of these results reads ... where $ If$ is the Hadamard finite part of $ ^b_af(x)\,dx$, $ C$ is Euler's constant and $ (z)$ is the Riemann Zeta function. We illustrate the results with an example.

@article{sidi2012eulermaclaurin, 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: ... 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 ... 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 ... 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. }, added-at = {2012-08-11T05:31:46.000+0200}, author = {Sidi, Avram}, biburl = {https://www.bibsonomy.org/bibtex/24358b791cb9d3ed6c1ec8b130cf21071/drmatusek}, doi = {10.1090/S0025-5718-2012-02597-X}, interhash = {f13848f4e9186445c3f44b879f500877}, intrahash = {4358b791cb9d3ed6c1ec8b130cf21071}, journal = {Mathematics of Computation}, keywords = {analysis asymptotic integral mathematics numerical series}, month = oct, number = 280, pages = {2159-2173}, timestamp = {2013-09-12T11:09:51.000+0200}, title = {Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities}, url = {http://dx.doi.org/10.1090/S0025-5718-2012-02597-X}, volume = 81, year = 2012 }