Essential Mathematical Methods for the Physics Sciences Problem 15.7
Show that if f(z) has a simple zero at z0 then 1/ f(z) has a residue of 1/f'(z0). Then use this information to evaluate:
∫ sinθ/(a- sinθ) dθ, where the integral goes from -∏ to ∏.
The book has further hints:
The unit circle has only one pole at z= i*a-i(a2-1)^1/2 and therefore has a residue of -i/2 *(a2 -1)-1/2.
The Attempt at a Solution
The first part is:
1/f(z) has a simple pole at z0 leaving the residue to be limz→z0 (z-z0) * 1/f(z) = 1/f'(z0) by definition of the derivative.
But using this information to do the integral is escaping me. Can anyone just help me understand how to use the hint. I don't actually need the whole step by step guide to the integral.