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## Homework Statement

Show that:

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].

## Homework Equations

f(z) = f(0) + Ʃb

_{n}(1/(z-a

_{n})+1/a

_{n}) (from n=1 to ∞) , where b

_{n}is the residue of f(z) at a

_{n}.

## 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!