$\Omega$ is a domain, $D(a, r) \subset \Omega$, and $u$ is a harmonic function on $\Omega$. I need to show that $$u(a) = \frac{1}{2\pi} \int _0 ^{2\pi} u(a + re^{it}) ~dt$$
it is easy to show it if u is holomorphic using Cauchy's equation and parametrizing, but here the function is harmonic... How can I use that?
Since $u$ is harmonic in we can write $u = \Re(f(z))$ where $f$ is holomorphic in $\Omega$.By Cauchy's Integral Formula for holomorphic functions, we have $$f(a) = \displaystyle\frac{1}{2\pi i}\displaystyle\int_{C}\displaystyle\frac{f(z)}{z-a}\ dz$$ Substituting $z=a + re^{i\theta}$ gives $$ f(a) = \displaystyle\frac{1}{2\pi i}\displaystyle\int_{0}^{2\pi}\displaystyle\frac{f(a + re^{i\theta})}{re^{i}\theta}rie^{i\theta}\ d\theta = \displaystyle\frac{1}{2\pi}\displaystyle\int_{0}^{2\pi}f(a + re^{i\theta})\ d\theta $$ Now take the real parts of both sides.