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I need to find the Fourier transform of the function $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{(x-\mu)^2}{2\sigma^2}), x, \mu \in \mathbb{R}, \sigma>0$ by using the Cauchy residue theorem. The Fourier transform will have the expression $$\frac{1}{\sqrt{2\pi\sigma^2}}\int_{\mathbb{R}} e^{-(x-\mu)^2/2\sigma^2 + itx}\, dx.$$ I do not see an isolated singularity except for $x \rightarrow \infty.$ Can somebody provide some hint or propose a solution ? Thanks.

A rural reader
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  • Related: https://math.stackexchange.com/q/270566/814040 ? – Adithya May 16 '23 at 15:00
  • +1 At least for me it would be very interesting to see how, as requested, the Cauchy residue theorem could be used for this. No poles involved, just contour construction. – A rural reader May 16 '23 at 15:38
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    @Aruralreader and OP consider the simpler case $\mu=0, \sigma=1$ and read this https://math.stackexchange.com/questions/545431/fourier-transform-of-exp-t2-using-contour-integration?rq=1 – Levon Minasian May 17 '23 at 12:43

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