The question is attached, and is from Rudin's "Principles of Mathematical Analysis". Essentially, the question is this:

Let f be a continuous complex function defined in the complex plane.

Suppose there is an integer n and a nonzero complex number such that lim z^(-n)f(z)=c

Prove that f(z)=0 for at least one complex number z

(the suggested three steps of the proof are included in the attached document)

It is suggested that we use the index/winding number of a curve to prove this.

Let f be a continuous complex function defined in the complex plane.

Suppose there is an integer n and a nonzero complex number such that lim z^(-n)f(z)=c

Prove that f(z)=0 for at least one complex number z

(the suggested three steps of the proof are included in the attached document)

It is suggested that we use the index/winding number of a curve to prove this.

#### Top Answer

Here is a detailed explanation... View the full answer