On the error term of the filon quadrature formulae

Abstract

Errors are noted and corrected in existing descriptions of the local error terms in the Filon quadrature formulae. Expressions are established valid for all values ofθ=qh whereh is the step length andq the angular frequency of the fundamental oscillatory component. It is shown that the commonly used simplified error terms are only valid on intervals ofθ, approximately separated byπ, whose lengths diminish like 3/θ asθ gets large. A closed form local error term for a generalised Filon formula is also determined.