Evaluation of Contour integral

**lllll** · December 1st 2008, 06:26 PM

Given:

$\int_C z(\overline{z}+1)^2 dz$

where $C$ is the circle $|z+1|=3$ in a clockwise direction.

I'm thinking that the answer is simple 0 since it's composite function seems to be analytic everywhere, but it's continuous nowhere, since:

$(x+iy)(x-iy+1)^2 = (x+iy)(x^2-2iyx-y^2+2x-2iy+1)$ $= x^3-iyx^2 +y^2x-iy^3+2x^2+2y^2+x+iy$ which clearly won't satisfy the Cauchy-Riemann equations

**ThePerfectHacker** · December 1st 2008, 06:42 PM

Originally Posted by lllll

Given:

$\int_C z(\overline{z}+1)^2 dz$

where $C$ is the circle $|z+1|=3$ in a clockwise direction.

You can easily do this problem directly by contour integration.
The function $g(\theta) = 1 + 3e^{i\theta}$ for $0\leq \theta \leq 2\pi$ is an image of this circle.
We also see that $\dot g (\theta) = 3ie^{i\theta}$ .

Thus, the integral becomes,
$\int \limits_0^{2\pi} (1+3e^{i\theta}) (3e^{-i\theta} + 2)^2 (3ie^{i\theta}) d\theta$

Open parathesis and get your answer.
Hint: $\int \limits_{0}^{2\pi} e^{ik\theta}d\theta = 0 \text{ if }k\in \mathbb{Z}^{\times}$

Thread: Evaluation of Contour integral

LinkBack

Thread Tools

Display

Evaluation of Contour integral

Similar Math Help Forum Discussions

contour integral, limiting contour theorem with residue

Evaluation of a real integral using Contour Integration

Integral Evaluation

Use Cauchy's integral theorem for derivatives to evaluate the contour integral.

evaluation of an integral

Search Tags