Real integrals with complex coefficients

[VantagePoint72]

I'm curious about the validity of various techniques from good old calculus in one real variable when dealing with complex coefficients. I know enough complex analysis to know that the rules change when dealing with complex variables, but I'm curious about the case when the variables are still real and its just coefficients that are complex.

For example, suppose you wanted to do the following integral:

$$\int_{-\infty}^{\infty} e^{-\left(t + i\alpha\right)^2} dt$$

with $\alpha$ real. Can I just do u-substitution like this:

$$u = t + i\alpha \rightarrow du = dt \\<br /> \int_{-\infty}^{\infty} e^{-\left(t + i\alpha\right)^2} dt =<br /> \int_{u=-\infty}^{u=\infty} e^{-u^2} du$$

and conclude the integral is $\sqrt{\pi}$ like usual? Punching it into Wolfram Alpha confirms that's the result, so it's more whether or not what I did was valid in general that I'm interested in, rather than this particular result. Can I still use the familiar tricks, or does the mere presence of complex numbers invalidate these old techniques?

 

[mathman]

I am not sure about the validity of the technique but in this case it looks like the right answer. Expand the exponent and get e^α2 multiplying the Fourier transform of e^-t2, which is e^-α2 (with some constants including √π).