For what values of m and n does

Saul Cochran

Saul Cochran

Answered question

2022-04-15

For what values of m and n does Czmzndz=0 and for what values does Czmzndz=2iπ?
I am stuck on this problem, any hint?

Answer & Explanation

Videoad3u

Videoad3u

Beginner2022-04-16Added 15 answers

Assuming C is a unit circle around origin we have this. Parametrize the circle as z=eit , t[0,2π]. then we have
Czmndz=02πeit(mn)(ieit)
Now if m-n+1=0, the integrand is constant and we get the result 2πi. Otherwise the integral vanishes.
Now, for general curve, the integral is only nonzero when m-n=-1. In that case, it measures what is called a winding number w, i.e. how many times does the contour wind around the origin, and the result is 2πiw. We had a special case of this in the first paragraph where w=1, since the circle wraps around the origin just once.

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