---
tag: SKE8QTW3YM
kind: concept
name: Peak function
msc: ["32E35", 32T15]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
---

## Summary

A peak function at a boundary point $p$ of a domain is a holomorphic function, continuous up to the boundary, that equals $1$ at $p$.
At every other point of the closure its modulus is less than $1$.
A point with a peak function is a peak point.
Peak functions help to localize problems at the boundary and to show that a domain is a domain of holomorphy.
They are also used to study the boundary behaviour of holomorphic maps.

## Explanations

### L1 human

A peak function is a holomorphic function whose modulus has a single highest point on the closed domain, and that point is on the boundary.
Powers $f^k$ of a peak function $f$ are then close to $1$ near $p$ and close to $0$ away from $p$, so they act like holomorphic "spotlights" on $p$.
On the closed unit disc, $f(z) = (1 + z)/2$ peaks at $z = 1$.
On the bidisc, the boundary points $(1, b)$ with $\abs{b} < 1$ have no peak function, because the boundary contains a flat disc through them.