# Complex Line Integral

Prove that if a differentiable curve $g:[0,1] \to \mathbb{C}$ (complex plane) parametrizes counterclockwise the boundary of an open set $O$ in $\mathbb{C}$, then under suitable conditions area of $O$ is $${1 \over 2i } \int_g \overline{z} dz$$ computed over the boundary.

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Write your contour integral out explicitly using the parameterization $g(t)$ and then use Green's theorem.
for example $\int(xx' + yy')$ is $\int(Pdx + Qdy)$ where $P$ is $x$ and $Q$ is $y$. ${\partial Q \over \partial x}$ and ${\partial P \over \partial y}$ are thus both zero and using Green's Theorem the integral of the first term is zero. Now try a similar thing in the second term and see what happens. Btw you don't even need the Cauchy-Riemann equations, I don't know why I said that. –  Zarrax Jan 23 '11 at 22:42