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I am sure this is quite a trivial problem, but i am stuck and was wondering if anyone could help.

I want to solve the integral:

$\displaystyle \dfrac{-m-n}{2\pi i}\int \dfrac{dz}{(1+z)}$ for a complex valued $z$ and $|z| < 1$.

I tried using radial coordinates but the result does not make too much sense...

$\displaystyle \dfrac{-(m+n)}{2\pi i}\int \dfrac{dz}{(1+z)} =\dfrac{-2\pi(m+n)}{2\pi i}\int^1 _0 \frac{z dz}{1+z}=\dfrac{-i(m+n)}{2}.$

thank you!

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I'm not sure I understand what path you're supposed to integrate on. –  Javier Badia Jul 7 '13 at 13:31
    
You need to specify a contour. –  user46080 Jul 7 '13 at 13:32
    
As far as I can tell, we are thinking of $\mathbb{C}$ as $\mathbb{R}^2$ and integrating $(1+x+iy)^{-1}$ over the unit circle $x^2 + y^2 < 1$? –  Cocopuffs Jul 7 '13 at 13:39
    
Since the domain $\{|z|<1\}$ is simply connected and since $\frac{1}{1+z}$ is holomorphic there, it admits holomorphic antiderivatives there, defined up to a constant: that's $\int \frac{1}{1+z}dz$. For instance $\int\frac{1}{1+z}dz=\log(1+z)+C$ where $\log$ denotes the principal branch of the complex logarithm. –  1015 Jul 7 '13 at 13:52
    
i don't know what contour to choose... this is all i have to go on... –  johanna Jul 7 '13 at 14:02

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