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I am trying to workout the following contour integral:

EDIT1: I am working on a simpler integral:

$$\oint_{C_{1,2}} dz \,\sqrt{\frac{z^2 +c^2 }{(z-z_1)(z-z_2)}}\, $$

on the small circles $C_{1,2}$ centered around the points $z_1$ and $z_2$ respectively. In the numerator $c$ is a constant.

How should we in general approach contour integrals (like the one above), where we want to calculate the contribution of poles given inside a square root ? Any suggestion is appreciated.

EDIT2: parametrization seems to work in such integrals. Using $z=z_2+re^{i\phi},\, \phi\, \in [-\pi, \pi]$ (not sure the interval is correct could be $[0, 2\pi]$), this is what I got:

$$\int^{\pi}_{-\pi} d\phi \,e^{i\phi/2}\sqrt{r}\sqrt{\frac{(z_2 + r e^{i\phi})^2 +c^2 }{(z_2-z_1+r e^{i\phi})}}\, $$

I am assuming I should take the limit $r\rightarrow 0$ at the end.

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    Why not $z$ insted of $r$ ? What is the contour ? The union of contours ? If yes how are oriented the circles ? – 2017-01-24
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    just a notation. No, each contour respectively. – 2017-01-24
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    Take the habit to use classical notations, with the classical dz at the end... – 2017-01-24
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    Try to use the following approach in your special case... https://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_4_.E2.80.93_branch_cuts – 2017-01-24

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