Why not use the residue theorem?
I'm stuck on a question of complex integration; there doesn't seem to be any way to integrate the function, so I imagine there must be a 'trick' somewhere, but I can't work it out Here's the problem:
Compute
around .
I parametrized the path as
but then I end up trying to integrate
which I can't work out. Thanks for any help
Well, you gotta make sure the numerator doesn't also go to zero or do something else equally strange at the same location. The precision necessary to correctly define a simple pole means you're probably going to be doing something similar to what your lecturers have. However, this is an intuitive way to think of poles, and it works in quite a few cases.
The numerator will be zero at that point, correct. However, the denominator has a 15th order zero at the origin. You can show that the sin function has a first-order zero at the origin. Therefore, the integrand has a 14th order pole at the origin. Make sense?
I would just evaluate the residue using this formula.