# Complex Contour Question

• Oct 25th 2009, 03:31 PM
Haven
Complex Contour Question
Let C be a positively oriented simple close contour. Show the area of region enclosed by C can be written as:

$\displaystyle \frac{1}{2i}\int_C z^{*} \mathrm dz$

Where $\displaystyle z^{*}$ is the conjugate of z.

I have no idea where to start on this problem. I was thinking maybe countour deformation, but i have no idea how to get the area.
• Oct 25th 2009, 04:00 PM
TheEmptySet
Quote:

Originally Posted by Haven
Let C be a positively oriented simple close contour. Show the area of region enclosed by C can be written as:

$\displaystyle \frac{1}{2i}\int_C z^{*} \mathrm dz$

Where $\displaystyle z^{*}$ is the conjugate of z.

I have no idea where to start on this problem. I was thinking maybe countour deformation, but i have no idea how to get the area.

This should get you started.

$\displaystyle z=x+iy \implies \bar{z} =x-iy$ and

$\displaystyle dz=dx+idy$

Write out the integral and use Green's theorem.