0
$\begingroup$

Any ideas about this integral . Prove that :
$ \mathop{\int}\limits_{L}{\frac{dz}{{z}\mathrm{{-}}{a}}}\mathrm{{=}}\mathrm{\pm}{2}\mathit{\pi}{i} $

If $ L $ is any closed rectifiable Jordan curve whose interior contains the point a ,where we choose the plus sign if $ L $ is traversed in the positive direction and the minus sign if $ L $ is traversed in the negative direction .

  • 0
    I am just thinking about the cauchy' s Integral theorem for a system of contours . – 2017-01-11

1 Answers 1

0

The theorem you are looking for states

Let f be a function continuous on the directed smooth curve $\gamma$. Then if $z = z(t)$, $a \leq t \leq b$ is any admissible parametrization of $\gamma$ consistent with its direction, we have

$$\int_\gamma f(z) dz = \int_a^b f(z(t))z'(t) dt$$

As a hint a suitable parametrization for L is $z(t)= a +re^{\pm i t}$

depending on the direction you integrate.