$\int_{c}\frac{e^{az}}{\cosh(z)}\:dz.$ for $c: |z|=2 $

Question

let $a$ be a real number and $z$ is a complex variable , I accrossed the bellow question in my textbook which i can't solving it, only it seems to me that this integral w'd be expressed as hypergeometric series , then my question is:

Question:How do i compute this integral over complex number which is defined as : $$\int_{c}\frac{e^{az}}{\cosh(z)}\:dz.$$ for $c: |z|=2 $ in the positive direction . ?

Note: I'm a bignner in integration over $\mathbb{C}$

Answer 1

By the residue theorem, this is equal to$$2\pi i\left(\operatorname{res}_{\frac{\pi i}2}\left(\frac{e^{az}}{\cosh z}\right)+\operatorname{res}_{-\frac{\pi i}2}\left(\frac{e^{az}}{\cosh z}\right)\right).$$These residues can be computed using the formula$$\operatorname{res}_a\left(\frac{f(z)}{g(z)}\right)=\frac{f(a)}{g'(a)}.$$