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Let $u(x, y)$ and $v(x, y)$ be differentiable real functions with continuous partial derivatives in a neighbourhood of $z_0$. Prove that $f = u+ iv$ is complex differentiable at $z_0$ if and only if $$\displaystyle\lim_{r\to 0}\frac{1}{\pi r^2}\oint_{C(z_0,r)}f(z)dz=0$$

Where $C(z_0, r)$ is the circle of radius r centered at $z_0$. I think this maybe related to Cauchy formula but the condition seems to point toward Cauchy Riemann equations.

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    Note that it doesn't say that $f$ is necessarily holomorphic at $z_0$, only that it is complex differentiable at $z_0$. The former would imply complex differentiable in some *neighbourhood* around $z_0$, which is *stronger*. This rules out the cauchy formula, I think, so you'll need to basically show that this limit only holds if the partial derivatives fulfill the Cauchy-Riemann differential equations at $z_0$.2012-10-24
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    I think i've figured it out, the main idea is to use Green's theorem to change the loop integral into Cauchy Riemann related expression. Then take the limit LOL2012-10-24

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