Question 7.29: Suppose f(z) satisfies the same conditions given in Problem ...
Suppose f(z) satisfies the same conditions given in Problem 7.25. Prove that
\sum\limits_{-\infty}^{\infty}(-1)^{n} f(n)=-\{\text { sum of residues of } \pi \csc \pi z f(z) \text { at the poles of } f(z)\}
We proceed in a manner similar to that in Problem 7.25. The poles of \csc \pi z are simple and occur at z=0, \pm 1, \pm 2, \ldots
Residue of \pi \csc \pi z f(z) at z=n, n=0, \pm 1, \pm 2, \ldots, is
\underset{z \rightarrow n}{\lim}(z-n) \pi \csc \pi z f(z)=\underset{z \rightarrow n}{\lim} \pi\left(\frac{z-n}{\sin \pi z}\right) f(z)=(-1)^{n} f(n)
By the residue theorem,
\oint\limits_{C_{N}} \pi \csc \pi z f(z) d z=\sum\limits_{n=-N}^{N}(-1)^{n} f(n)+S (1)
where S is the sum of the residues of \pi \csc \pi z f(z) at the poles of f(z).
Letting N \rightarrow \infty, the integral on the left of (1) approaches zero (Problem 7.106) so that, as required, (1) becomes
\sum\limits_{-\infty}^{\infty}(-1)^{n} f(n)=-S (2)