Adjusted Forms of the Fourier Coefficient Asymptotic Expansion and Applications in Numerical Quadrature

The conventional Fourier coefficient asymptotic expansion is derived by means of a specific contour integration. An adjusted expansion is obtained by deforming this contour. A corresponding adjustment to the Euler-Maclaurin expansion exists. The effect of this adjustment in the error functional for a general quadrature rule is investigated. It is the same as the effect of subtracting out a pair of complex poles from the integrand, using an unconventional subtraction function. In certain applications, the use of this subtraction function is of practical value. An incidental result is a direct proof of Erdélyi's formula for the Fourier coefficient asymptotic expansion, valid when f(x) has algebraic or logarithmic singularities, but is otherwise analytic. 1. The Fourier Coefficient Asymptotic Expansion. The Fourier coefficient asymptotic expansion (F.C.A.E.) (1.3) below is a classical formula which is elementary to derive using a standard application of the formula for integration by parts, namely, dx. /i i\ Í u ^ <*• -v e' f(b) — g' 7(g) 1 f ,„, ikx (1.1) J j(x)e dx =-— J f(x)e The integral on the right is almost the same as that on the left; the only difference is that f(x) replaces j(x). Consequently, the formula may be applied iteratively. So long as (1.2) Kx)ECM[a,b], this leads to the following series: f* Kx)e'kl dx = -e (1 -3) + e"'\i /(a) + p f(a) +...+£ /'"""(a)}