a tough integral
I have this problem:
It's like a two parts problem.
The first one was to prove the equation in red, I did that with a bit of trigo.
The second part is to use the integral of the function in green to prove the thing in yellow. I'm stuck on the path that goes from - epsilon to epsilon, that semi circle, how can I find the value of this function on the path?
When x is small, sin(x)/x is close to 1. So you are integrating a function that is approximately 1 over a very short interval (of length πε), and the integral will be small, tending to 0 as ε→0. Another way to see this is to notice that the singularity at 0 is removable, so what happens near it does not really affect the integral.