An old complex analysis exam question:
Evaluate
$$\large I(a) = \int_{-\infty}^{\infty} e^{-\frac{1}{2}x^2+iax}dx$$
So far, I have completed the square in exponent, and now I have the integral
$$\large I(a) = \int_{-\infty}^{\infty} e^{(x+\frac{ia}{2})^2-\frac{3}{2}x^2+\frac{a^2}{4}}dx$$
Perhaps completing the square above gives me an integral where I now don't have to worry about whether a is positive or negative -- and so I don't need to break the integration out into two cases.
But I'm not sure what to do from here. Most integration that I come across in old exam questions is an application of the Residue Theorem, but currently the integrand looks pole-free / entire, so there'd be no residues to compute.
One last thing I thought of so far is that the integral looks a bit like the Gaussian integrals. Would this be the better / correct path to follow? If so, what should I start with? The $iax$ term makes it tricky to know what sort of substitution I could go with to perhaps get something like $e^{-ax^2}$
Any hints or suggestions are welcome.
Thanks,