Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This problem is from Section VIII.6 of Theodore Gamelin's Complex Analysis:

Let $f(z)$ be analytic on an open set containing a closed path $\gamma$, and suppose $f(z)\neq0$ on $\gamma$. Show that the increase in arg$f(z)$ around $\gamma$ is $2\pi W(f\circ\gamma,\, 0)$.

Any hints?

share|cite|improve this question
Hint: The definition of the winding number is $W(f\circ\gamma) = \frac{1}{2\pi i}\int_{f\circ \gamma}\frac{dz}{z}$. The differential $\frac{dz}{z}$ is exactly $d\log z$ (why does this make sense?), so $W(f\circ \gamma) = \frac{1}{2\pi i}\int_{f\circ \gamma}d\log z$. The integral is exactly the change in $\log z$ as you traverse $f\circ \gamma$ (why?). Finally, what is the change in $\log z$ as you traverse $f\circ \gamma$? – froggie Nov 27 '12 at 13:58

Your Answer


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

Browse other questions tagged or ask your own question.