The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. III. Functions having algebraic singularities

Abstract

The purpose of this paper is to extend the MIPS theory described in Parts I and II to functions having algebraic singularities. As in the simpler cases, the theory is based on expressing the remainder term in the appropriate Fourier coefficient asymptotic expansion as an infinite series, each element of which is a remainder in the Euler-Maclaurin summation formula. In this way, an expression is found for ∫ a b f ( x ) cos ⁡ 2 π m x d x ( 0 ≦ a > b ≦ 1 ) \smallint _a^bf(x)\cos 2\pi mx\;dx\;(0 \leqq a > b \leqq 1) where f ( x ) = ( x − a ) α ( b − x ) β ϕ ( x ) , ϕ ( x ) f(x) = {(x - a)^\alpha }{(b - x)^\beta }\phi (x),\phi (x) being analytic and α , β > − 1 \alpha ,\beta > - 1 . This expression, and variants of it form a convenient basis for the numerical calculation of a set of Fourier coefficients. The calculation requires approximate values of the first few derivatives of ϕ ( x ) at x = a \phi (x)\;{\text {at}}\;x = a and at x = b x = b , together with trapezoidal rule sums over [0, 1] of f ( x ) f(x) . Some of the incidental constants are values of the generalized zeta function ζ ( s , a ) \zeta (s,a) .