I am interested in evaluating a Fourier transform integral $$ \int_{0}^{\infty} f(x) \sin(kx) dx $$ where $$ f(x) = \frac{\exp(-a x^2) x}{x^2 + b^2} $$ with $a, b, k > 0$. In Erdelyi's Tables of Integral Transforms, Vol. 1, pp. 74, equation (26), this integral evaluates to the following: $$ \frac{\pi}{4} \exp(a^2 b^2) [\exp(-bk) \, \text{erfc}(ab - \frac{k}{2a}) - \exp(bk) \, \text{erfc}(ab + \frac{k}{2a})] $$ But I cannot reproduce this result using contour integration. Can this integral be evaluated using contour integration? If not, is there a way to evaluate it directly?
2026-05-14 08:50:56.1778748656
Evaluation of a Fourier transform integral
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