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For a small problem I'm working on in quantum field theory, I have to numerically evaluate the integral

[tex]\int_{-\infty}^{t}du e^{i\omega u}erf\left(\frac{u}{\sqrt{2}\sigma}\right)[/tex]

where [itex]erf(z)[/itex] is the error function.

Now, I have to replace [itex]-\infty[/itex] by some large negative number, so I effectively end up evaluating

[tex]\int_{-t_H}^{t}du e^{i\omega u}erf\left(\frac{u}{\sqrt{2}\sigma}\right)[/tex]

If I use Matlab or Mathematica to evaluate this integral numerically, I get warnings indicating that the integrand is singular, or highly oscillatory. Secondly, the choice of [itex]t_H[/itex] seems critical, and since the integrand itself doesn't fall off asymptotically, its not clear to me how [itex]t_H[/itex] should be chosen in terms of [itex]\sigma[/itex].

Note that in the [itex]t_H, t \rightarrow \infty[/itex] limit, this is just the Fourier Transform of the error function, which is well defined for [itex]\omega \neq 0[/itex]. Incidentally, [itex]\omega \neq 0[/itex] is ensured in my physical problem.

Any suggestions for how this integral could be numerically evaluated?

Secondly, what is a good way to evaluate the error function of a complex number, i.e. erf(a + ib) where a and b are real, and i = sqrt(-1)?

Thanks in advance!

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# Problem with numerical integration of error function

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