Answer to Question #258276 in Complex Analysis for yuvasri

Question

4. What is the value of

sum - infty ^ infty 1/{ (z+n)^ 2 +a^2}?

Answer 1

$\displaystyle{\sum_{-\infin}^{\infin}}\frac{1}{(z+n)^2+a^2}=2\pi i\cdot \sum res\ f(z_0)$ for $z_0$ in the upper half-plane

 $z_0=\pm ia-n$

$res_1\ f(z_0)=\displaystyle{\lim_{z\to z_0}}[f(z)(z-z_0)]=\frac{1}{2ia}$

$res_2\ f(z_0)=-\frac{1}{2ia}$ is not in the upper half-plane if a > 0

then:

$\displaystyle{\sum_{-\infin}^{\infin}}\frac{1}{(z+n)^2+a^2}=2\pi i/2ia=\pi/a$