Herglotz–Riesz kernel

Remarks

For fixed $\zeta$, the function $z \mapsto H(\zeta, z)$ is holomorphic on $D$. Its real part is the Poisson kernel: with $a = \zeta - c$ and $b = z - c$, $$\operatorname{Re} H(\zeta, z) = \operatorname{Re} \frac{(a + b)(\bar a - \bar b)}{\abs{a - b}^2} = \frac{R^2 - \abs{z - c}^2}{\abs{\zeta - z}^2},$$ because $b \bar a - a \bar b$ is purely imaginary. Example: for the unit disc ($c = 0$, $R = 1$), $\zeta = 1$ and $z = 1/2$, one gets $H = 3$, and the Poisson kernel $(1 - 1/4)/(1/4)$ is also $3$. Non-example: the Cauchy kernel $\zeta/(\zeta - z)$ is also holomorphic in $z$, but at $\zeta = 1$, $z = 1/2$ it equals $2$, so its real part is not the Poisson kernel. For $c = 0$ the two kernels are related by $H(\zeta, z) = 2\zeta/(\zeta - z) - 1$. If $u$ is harmonic on a neighbourhood of $\overline{D}$, the Schwarz integral formula reads $$u(z) + i v(z) = \frac{1}{2\pi} \int_0^{2\pi} H(c + R e^{i\theta}, z) \, u(c + R e^{i\theta}) \, d\theta,$$ where $v$ is the harmonic conjugate of $u$ on $D$ with $v(c) = 0$. The same kernel appears in the Herglotz representation theorem. Other names in use are Herglotz kernel and Schwarz kernel.