Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective
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In this note, we provide a new perspective on Euler–Maclaurin expansions of (offset) trapezoidal rule approximations of the finite-range integrals I[f]=∫baf(x),dx, where f ∈ C∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms Open image in new window where Ps(y) and Qs(y) are some polynomials in y. Here the γs and δs are complex in general and different from −1,−2,... . The results we obtain in this work generalize, and include as special cases, those pertaining to the known special cases in which f(x)=(x−a)γ[ log (x−a)]pga(x)=(b−x)δ[log (b−x)]qgb(x), where p and q are nonnegative integers and ga ∈ C∞[a,b) and gb ∈ C∞(a,b]. In addition, they have the pleasant feature that they are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. With h=(b−a)/n, where n is a positive integer, and with Open image in new window one of these results reads, as h→0,Open image in new window where ζ(z) is the Riemann Zeta function.
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