I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I can't find any references which solve similar problems.
In my case, I need to integrate a contour around the unit circle where the denominator contains the fourth root of a polynomial of degree $12$. It happens that $6$ of the $12$ roots are inside the unit circle and have a known value. I also know the value of the other $6$ roots outside of the unit circle but I do not think they are as important.
Does anyone have a source that will help me deal with this problem? Can anyone sketch a method of calculating this? I have tried integrating from one root to it's conjugate assuming the other sections of the contour would have no contribution but I have not received correct results.
Can I assume that there is no contribution from the circles that go around the zeros of the polynomial?