Interesting complex variables problem

[nicksauce]

Homework Statement

Let $$\Omega$$ be a bounded domain in C whose boundary is a curve z = z(t), a<=t<=b, and let $$A(\Omega)$$ be the area of $$\Omega$$. Prove that

$$A(\Omega) = \frac{1}{2}\int^b_a |z(t)|^2 Im(\frac{z'(t)}{z(t)})dt$$

Homework Equations

The Attempt at a Solution

Not even sure where to start on this. Any tips?

 

[Dick]

Yes. $$A(\Omega) = \frac{1}{2}\int_C x dy-y dx$$. That's a well known expression for calculating area using Green's theorem. If you express z(t)=x(t)+iy(t), that's what your expression reduces to.

 

[nicksauce]

Ahh right, thank you