Euler–Maclaurin summation and Schlömilch series
نویسنده
چکیده
A method for analysing a class of divergent series is developed from the Euler– Maclaurin summation formula. The conditions that the summand must satisfy are explored, and a significant simplification is obtained for cases where the summation ranges over all integers. As an example, we consider the Ewald representation for Schlömilch series, and show that this includes Twersky’s dual series formula as a limiting case.
منابع مشابه
Resurgence of the Euler-MacLaurin summation formula
Abstract. The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Pla...
متن کاملAn Accurate Estimate of Mathieu’s Series
Using Hermite’s, i.e. the Euler-Maclaurin summation formula of order four, new approximations to Mathieu’s series S(x) ≡ ∑∞ k=1 2k (k2+x2) are obtained, which are more accurate than the approximations presented recently in the literature. Mathematics Subject Classification: 26D15, 33E20, 33F05, 40A05, 40A25, 65B10, 65B15, 65D20
متن کاملThe Euler-Maclaurin Formula and Sums of Powers Revisited
Using the Euler-Maclaurin summation formula the strictly increasing convergence lim m → ∞ m j=1 j m m = e e − 1 is demonstrated.
متن کاملThe Calculation of Fourier Coefficients
The Möbius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis...
متن کاملNotes on Euler-boole Summation
We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases in a general approach to approximation that also encompasses Taylor sums. Here we give additional details of the construction.
متن کامل