= Poisson formula for a holomorphic function
{c}
{title2=$f(z)=\frac1{2\pi}\int_0^{2\pi}f(Re^{i\theta})\operatorname{Re}\frac{Re^{i\theta}+z}{Re^{i\theta}-z}d\theta$}
For $f$ holomorphic on a neighbourhood of the radius-$R$ <closed disk> and $|z|<R$, the displayed representation holds even though $f$ may be complex-valued. For $z\ne0$, put $w=R^2/\bar z$. On $|\zeta|=R$, the <Poisson kernel> equals $\zeta/(\zeta-z)-\zeta/(\zeta-w)$. Replacing $d\theta$ by $d\zeta/(i\zeta)$ reduces the formula to the <Cauchy integral formula> minus a zero contour integral whose pole lies outside. The case $z=0$ is the <mean value property for holomorphic functions>.
Back to article page
