I am wondering whether the following integral
$\int_{-\infty}^{\infty} \exp( - a x^2 ) sin( bx ) / x$
exists in closed form. I would like to use it for numerical calculation and find an efficient way to evaluate it. If analytical form does not exist, I really appreciate any alternative means for evaluating the integral. One method would be numerical quadrature including Gaussian quadrature, but it may be inefficient when the parameters a
and b
are very different in scale.
thanks in advance