# Derivation of the Poisson Kernel from the Cauchy Formula

Let $f$ be a holomorphic function on the disk

$$D_{r_0} =\{z\in\ {C} : |z|<R_0 \}$$ centered at the origin and of radius $R_0$.  Prove that whenever $0<R<R_0$ and $|z|<R$, then $$f(z) =\frac{1}{2π}\ \int_{0}^{2\pi} f(Re^{i\phi}) Re \bigl ( \frac {Re^{i\phi}+z}{Re^{i\phi}-z} \bigr) d \phi.$$ and the lecture note start with below

$$f(z) =\frac{1}{2πi}\ \int_{|ζ|=R} \frac{f(ζ)}{ζ−z}\ dζ.$$

$$0 =\frac{1}{2πi}\ \int_{|ζ|=R} \frac{f(ζ)}{ ζ−\frac{R^2}{\bar{z}} }\ dζ.$$

My lecture note says that the second equation holds by Cauchy Theorem. But I don't know why the second equation is equal to zero.$\frac{R^2}{\bar{z}}$ could be on the disk which means ${|ζ|=R}$. Am I wrong?.

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Where does $D_{R_0}$ appear, in your lecture notes? It seems you never use it. – Siminore Oct 4 '12 at 9:49
I don't exactly see how the term $\frac{R^2}{\bar{z}}$ comes up. More specifically, is the second equation supposed to hold for any choice of $z\in\mathbb{C}$? – Giovanni De Gaetano Oct 4 '12 at 9:50

The second equation hods for $|z|<R$. I assume that in the third equation we also have $|z|<R$. Then $|R^2/\bar z|>R$ and the function $$\frac{f(z)}{z-R^2/\bar z}$$ is holomorphic on an open set containing $\{|z|\le R\}$. Cauchy's Theorem implies that the integral is zero.