(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Calculate the closed path integral of [tex]\frac{z+2 i}{z^3+4 z}[/tex] over a square with vertices (-1-i), (1,i)

and so forth.

2. Relevant equations

The closed line integral over an analytic function is 0

3. The attempt at a solution

Alright, so first I factored some stuff, leaving me with

[tex]\frac{1}{z (z-2 i)}[/tex]

Now, this has a simple pole at 0 and at 2i. However, 2i isn't in the square, so I'm only concerned about the one at 0. Now, I don't know how to continue from here on, formally. I know (do I?) that the answer has to be i*2pi, as that always happens to be the case when doing a closed integral of a function that has 1 singularity (and it is enclosed by the path). But how do I 'calculate' this? I tried writing z = x+iy and the same for dz, and then working it all out and parameterizing the vertices, but that gives horrible expressions to integrate and just doesn't seem right.

Is there something I'm missing?

Also, there is a follow up question, which concerns the same integral but now over the square with vertices (-1-3i), (1+3i) and so forth. So this encloses both singularities, but they aren't symmetric or anything, so then I can't really go any further. (Is it even true, that when the singularities are distributed symmetrically, that they sort of cancel each others effect and your integral evaluates as 0?)

Kind regards

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# Cauchy-Goursat's theorem and singularities

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