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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.

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up vote 2 down vote accepted

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 $C$.

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Oh, that's very obvious, thanks! – Argon Mar 15 '12 at 11:37

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