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I am reading through Bak and Newman's Complex Analysis, where typically the authors are pretty telegraphic in their proofs. However, I am a little confused by their proof of MVT for complex numbers, and I believe my difficulty is just in the algebra/real-valued calc.

I can't get from the step

$f(a) = \frac{1}{2\pi i}\int_0^{2 \pi}{\frac{f(a+re^{i \theta})}{re^{i \theta}}d \theta}$

to

$f(a) = \frac{1}{2\pi}\int_0^{2 \pi}{f(a+re^{i \theta})d \theta}$

I've tried substituting $u$ for $re^{i \theta}$ but that doesn't seem to get me anywhere.

What is the algebra to make this true?

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Your first equation is not a correct rewriting of Cauchy's formula, if that's what you're trying to do. Specifically, you forgot to compute dz. –  Potato Oct 8 '12 at 1:45
    
@Potato, thank you. $dz$ != $d \theta$ of course. –  tacos_tacos_tacos Oct 8 '12 at 1:47
    
@Potato seriously thank you... I spent > 20 minutes trying to "solve" the integral by reversing the product rule for derivatives, which is an exercise in pain and futility... –  tacos_tacos_tacos Oct 8 '12 at 1:49

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