Using residue theorem, calculate
How do I find the poles of this function?
Quite frankly, the fact that you keep writing " " indicates you need to brush up on "complex numbers" before trying contour integrals and residues.
i can be written in "polar form" as . It's fourth roots are given by which gives different result for k= 0, 1, 2, and 3.
With k= 0, a fourth root of i is . With k= 1, another is . With k= 2, . Finally, with k= 3, .
Two of those four roots have positive real part, two negative real part. Two have positive imaginary part, two negative imaginary part. That means you really only need to look at the residues at two poles, depending on how you set up the contour.