I came to this question by looking at the fourier transform of a hyperbolic cosine. Let's say $f(x)=\cosh(ax)$, where $a$ is a complex number and $x$ is real. Then the fourier transform is $F(\omega)=\sqrt \frac{\pi}{2} \delta(\omega-ia)+\sqrt \frac{\pi}{2} \delta(\omega+ia)$. So it's the sum of two dirac deltas that take their nonzero values at the complex numbers $z=\pm ia$. I want to know what happens if I integrate along a contour that surrounds one of these numbers (i.e. Does that dirac delta function have residue?)
I found a few references to this question online but so far have not come to a definitive answer. This is the best source I've found so far, but it doesn't seem to answer the question. He seems to integrate along a contour, where the dirac delta actually lies on the contour. So that's not really a residue. I guess this question is related to how to take the inverse fourier transform and recover the hyperbolic cosine, since you would need to integrate along a contour that contains the nonzero values of the dirac delta. Anyone have ideas?