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- Jun 20, 2014

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Let $\Bbb D$ be the open unit disc in the complex plane, and let $f$ be a continuous complex function on $\partial\Bbb D$. Consider the function

$$F(re^{i\phi}) \,\dot{=}\, \frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})\frac{1-r^2}{1-2r\cos(\theta-\phi) + r^2}\, d\theta\quad (re^{i\phi}\in \Bbb D)$$

Prove $F$ is harmonic on $\Bbb D$, and that for all $z_0\in \partial \Bbb D$, $\lim\limits_{z\to z_0} F(z) = f(z_0)$.

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