Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In topology, the winding number is homotopy invariant under the definition $n(\gamma,a)=\frac{\tilde{\theta}(\beta)-\tilde{\theta}(\alpha)}{2\pi}.$

I assume the must be true in the framework of complex analysis. Suppose you take as definition for the winding number $n(C,a)$ of a curve $C$ through $a$ to be $$ n(C,a)=\frac{1}{2\pi i}\int_C\frac{dz}{z-a}. $$

Is is still true that $n(C,a)$ is hopotopy invariant under smooth curves $C$ not going through $a$? Thank you.

share|improve this question
    
It seems intuitively that it would be the case that $n(C,a)$ is homotopy invariant under $C$, but I don't know how to prove it. –  Samuel Reid Feb 25 '12 at 7:12

1 Answer 1

up vote 1 down vote accepted

Both definitions agree, so the winding number is homotopy invariant. See Stewart and Tall, "Complex Analysis", section 7.5. You might be also interested in this question and, perhaps, my answer therein.

share|improve this answer
    
Thanks, is it possible to show it without referring back to the topological definition? –  Dedede Feb 27 '12 at 11:00
    
Indeed: look at op.cit., theorems 9.3 and 9.4. –  a.r. Feb 27 '12 at 13:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.