A robust double exponential formula for Fourier-type integrals

A double exponential transformation is presented to obtain a quadrature formula for Fourier-type integrals ∫0∞f(x)sin ωx dx or ∫0∞f(x)cos ωx dx where f(x) is a slowly decaying analytic function on (0,∞). It is an improved version of what we previously proposed in 1991. The transformation x=φ(t) is such that it maps the interval (0,∞) onto (−∞,∞), and that, while the integrand after the transformation decreases double exponentially at large negative t, the points of the formula approaches to zeros of sin ωx or cos ωx double exponentially at large positive t. Then the trapezoidal formula with an equal mesh size is applied to the integral over (−∞,∞) after the transformation, which gives an efficient quadrature formula for the Fourier-type integrals. The present transformation is improved in the sense that it can integrate a function f(z) with singularities in the finite z-plane more efficiently than the one previously proposed.