Ʃ(-1)n/(n^2+a^2) (from n=0 to ∞) = pi/[asinh(pi*a)], a[itex]\neq in[/itex], n[itex]\in Z[/itex].
f(z) = f(0) + Ʃbn(1/(z-an)+1/an) (from n=1 to ∞) , where bn is the residue of f(z) at an.
The Attempt at a Solution
The main problem is I don't how to pick the function f(z) to start with. I am thinking is pi*cot(pi*z) a good starting point? But in this case, we are dealing with the hyperbolic function, so coth will be a better choice??
Any help is appreciated!