---
tag: 1FSFX18D68
kind: concept
name: Herglotz–Riesz kernel
msc: [31A10, 30C40]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_harmonic_function
---

## Summary

The Herglotz–Riesz kernel of a disc is the holomorphic function of $z$ whose real part is the Poisson kernel.
Integrating it against the boundary values of a harmonic function $u$ gives a holomorphic function with real part $u$.
It is the kernel in the Schwarz integral formula and in the Herglotz representation of holomorphic functions with positive real part.

## Explanations

### L1 human

The Poisson kernel tells how much each boundary point of a disc counts when one rebuilds a harmonic function inside.
The Herglotz–Riesz kernel carries the same information, but as a holomorphic function.
For the unit disc and a boundary point $\zeta$ it is $(\zeta + z)/(\zeta - z)$.
At the centre $z = 0$ it equals $1$, and it blows up as $z$ approaches $\zeta$.