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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?

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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
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