Whittaker and Watson Theory of Residues example one

TriggerFish

Hi all,

I have been struggling with this exercise as well as number 4.
I was wondering if there is a solution / hint manual for this well known text available somewhere ?
Otherwise I would very much appreciate any hints on this one to start with
1. The problem statement, all variables and given/known data
-See attached image, more legible this way.

3. The attempt at a solution
Transform into a countour integral by substituting z = exp(iθ), replace sin(θ) and cos(θ)
by the obvious.
This gives a rational function of z, the poles are z=0, z=x, z=1/x ( x = 0 is excluded, that particular solution is trivial )
However having f(0.5(z+1/z), 0.5/i(z-1/z)) doesn't seem coherent with the definition given, and also calculating the residue for z=0 is problematic.
Another attempt would be to substitute with f(Re(z), Im(z)), however since all poles are real this would suggest all the residues are null given that f is null for real numbers !?

The passage from f to Phi also seems problematic, I thought the analicity of the function and two boundary conditions given would suffice to deduct something interesting using Cauchy's formula for partial derivatives of analytical functions but I wasn't able to.

Thank you very much in advance!

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