Numerical integration of oscillatory functions over Infinite interval by Integration by substitution using Taylor series

Abstract

Abstract. We propose a method to calculate the infinite interval oscillatory function integral $\int_0^\infty f(x)g(h(x))dx$ when $g(x)$ is an oscillatory function. By dividing $x=a$ into two and evaluating the integral over $[0,a]$ using numerical integration, the remainder is transformed to $t=h(x)$ and given the form $\int_{h(a)}^\infty F(t)g(t)dt$, which can then be expanded to an asymptotic series up to any order using partial integration and Taylor expansion. Calculations can be performed efficiently using this asymptotic expansion.