Evaluate $\displaystyle \int_{C(0,e)}\frac{1-cosz}{(e^z-1)sinz}dz$.
This function has an isolated singularity at $\displaystyle z=0$.
I tried this so many times, but still can't get it. Can I get some help please?
When z = 0, the function $\displaystyle \frac{1-cosz}{(e^z-1)sinz}$ has a zero of order 2 in the numerator, and also a zero of order 2 in the denominator. Therefore this singularity is removable. There are no other singularities inside the contour, so the integral must be 0.