Poisson formula for a holomorphic function ($f(z)=\frac{1}{2\pi }{\int }_0^{2\pi }f(R{e}^{i\theta })\mathrm{Re}\frac{R{e}^{i\theta }+z}{R{e}^{i\theta }-z}d\theta$)

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 $z\ne 0$, 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.