A robust double exponential formula for Fourier-type integrals

✍ Scribed by Takuya Ooura; Masatake Mori

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 170 KB
• Volume: 112
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00223-x

✦ Synopsis

A double exponential transformation is presented to obtain a quadrature formula for Fourier-type integrals

is a slowly decaying analytic function on (0; ∞). It is an improved version of what we previously proposed in 1991. The transformation x = (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 !x or cos !x 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 e cient 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 ÿnite z-plane more e ciently than the one previously proposed.