Why does $\oint_C d\log|f(z)|=0$? In the article on the argument principle on PlanetMath, it says that
$$\oint_C d\log|f(z)|=0$$ since $|f(z)|$ is single-valued. Why does that follow, or can someone point me to a fuller explanation? I'm studying complex-analysis right now, but this result is not obvious to me. Thanks.

Since $\log|f(z)|$ is single-valued, we have
$$
\oint_Cd\log|f(z)|=\log|f(z)|\bigg|_A^A=0
$$
where $A$ is any point on $C$.
