I have a question about Lebesgue integral with complex podintegral function (I am calculating the Fourier transform of the function $f(x)=e^{-x^2 } $ ). I tried following:
$$ \hat{f}(t) = \int_{\mathbb{R} } e^{-x^2 } e^{-itx}dx = \int_{\mathbb{R} } e^{ -( \frac{x}{\sqrt{2}} + \frac{i t}{\sqrt{2}} )^2 - \frac{t^2}{2}} dx = e^{- \frac{t^2}{2}} \int_{\mathbb{R} } e^{ -( \frac{x}{\sqrt{2}} + \frac{i t}{\sqrt{2}} )^2} dx $$
Now, is it allowed to use substitution $ \frac{x}{\sqrt{2}} + \frac{i t}{\sqrt{2}} = u$? Then, $dt = \sqrt{2}du$, but what happens to the set of integration i.e. limits of integral? When I solved this way, I got the result $ \sqrt{2 \pi} e^{ -\frac{t^2}{2}} $, but I am not sure if I can use the substition above in case of complex numbers.
I know there are some other methods to calculate this integral, but I am interested in correctness of this method. Thanks a lot in advance.