Residue Theory for Solving Complex Contour Integrals | Step-by-Step Guide

[everjordan]

Homework Statement

\oint (z-i)^2 \div sin^2(z)
gamma : 4+4i, 4-4i, -4+4i, -4-4i rectangluar

Homework Equations

residue theory

The Attempt at a Solution

I don't know how to solve this problem.
I do my best using taylor series... however it doesn't solve the problem..
Please help me...

 

[hgfalling]

Expand 1/sin z in its Laurent series and after multiplying the various terms together, pull the residues off the 1/z term.

Hint: what order pole does 1/sin z have? what's the residue of 1/sin z at 0?

 

[everjordan]

thank you for reply :)

first, i expanded csc^2(z) in Laurent series

and (z-i)^2 = z^2 -2iz -1

after that, multiply Laurent series and z^2-2iz-1

and pull the residues off the z^-1 term.

but I wasn't sure this was a good idea.

Is this right?

 

[hgfalling]

Now you just need to do the same thing for the other poles inside your contour, except you have to expand the series around those points instead. (so the series will be in (z-pi) and (z+pi) instead of just z).