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.

  • $\begingroup$ Why not $z$ insted of $r$ ? What is the contour ? The union of contours ? If yes how are oriented the circles ? $\endgroup$
    – Jean Marie
    Jan 24, 2017 at 7:46
  • $\begingroup$ just a notation. No, each contour respectively. $\endgroup$
    – user91411
    Jan 24, 2017 at 7:48
  • $\begingroup$ Take the habit to use classical notations, with the classical dz at the end... $\endgroup$
    – Jean Marie
    Jan 24, 2017 at 7:50
  • $\begingroup$ Try to use the following approach in your special case... en.wikipedia.org/wiki/… $\endgroup$
    – b00n heT
    Jan 24, 2017 at 8:46


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