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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

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    Oops, I meant the integral to be \int_{-\infty}^{\infty} \exp( - a (x - x0) ^2 ) sin( bx ) / x, where a, b, and x0 are constant parameters. But the integral in the above post is also helpful to me. Thanks!2012-10-06
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    Have you considered Fourier transforms? This is a convolution in physical space which is multiplication in Fourier space.2012-10-06
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    Following your comment, I have just tried Fourier transform and it looks like the integral can be written as a sum of error functions having complex arguments. My (very very) tentative result is like the integral = const x { erf( ( b - 2*i*a*x0 ) / 2\sqrt{a} ) - similar term }. So, if I can find a reliable math library routine for complex error function, I may be able to evaluate it efficiently.. I'll try along this line. Thanks :D2012-10-06

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