I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have
$$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I think I have to use the Residue theorem. (Note it has been years since I have had to do complex integration so I am winging it here).
Hence I must evaluate
$$\int_{-\infty}^\infty \frac{\sinh{w(a-b)}}{w \cosh{wa}}e^{-iwt}dw$$
which has multiple simple poles, one at $$w=0$$ and one at each n
$$w=(n+1/2)i\pi/a$$.
From here I am a bit lost. I think I need to create an integration contour which surrounds all (or is it one at a time?) of the poles. I would greatly appreciate any help. This is NOT a homework problem so you are not helping me cheat! It is a problem I have been working on for interests sake.
Thanks all