Strange complex analysis identity I recently had a homework problem in complex analysis where we had to prove the following identity: Let $f$ be a holomorphic function on a disk of radius $R_0$ centered at the origin. Then for $\vert z \vert < R$, where $0 < R < R_0$, we have
$$f(z) = \frac{1}{2\pi}\int_{0}^{2\pi}f(Re^{i\theta})\textrm{Re}\left(\frac{Re^{i\theta} + z}{Re^{i\theta} - z}\right)d\theta.$$
Does anyone know where this came from/if it has a name/etc.? Also, I'm curious about applications/when this integral can be evaluted/etc. I can't really envision a situation in which this would be useful.

This is the Poisson integral formula. It expresses the values of a harmonic function (in particular: of an analytic function) in the interior of the disk $D_R$ in terms of its values on the boundary $\partial D_R$. For the proof you start with the Cauchy integral formula and apply some symmetry considerations. Note that in the case at hand the function $f$ was analytic in $\bar D_R$ to start with. If the given function is just an arbitrary real function on $\partial D_R$ the integral gives you the (uniquely determined)  harmonic extension of $f$ into the interior of $D_R$.
