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The derivation of the Poisson kernel for a disc seems to involve a trick, and I don't really understand how one would come up with it.

Let $f$ be a holomorphic function on a disc $D_{R_0}$ centered at the origin and of radius $R_0$.

We are aiming for some sort of integral representation of $f$ with a real-valued kernel, which ends up to be

$$f(z) = \frac{1}{2\pi}\int_0^{2\pi} f(Re^{i\varphi}) \text{Re}\left(\frac{Re^{\varphi} + z}{Re^{i\varphi} - z}\right) \; d\varphi$$

So the trick used is to write

$$f(z) = \frac{1}{2\pi i} \int_{D_{R_0}} f(\zeta)/(\zeta-z) - f(\zeta)/(\zeta-R^2/\bar{z}) \; d\zeta$$

I'm not sure how one would come up with adding $f(\zeta)/(\zeta-R^2/\bar{z})$, except for the fact that it magically seems to work. I've tried working backwards, but it really hasn't given me any new insight into the problem.

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