---
tag: KGJD4A4ATY
kind: concept
name: Outer function
msc: [30H10, 30J99]
sources:
  - url: https://en.wikipedia.org/wiki/Hardy_space
---

## Summary

An outer function is a special holomorphic function on the [[unit disc|8G5ABXRK85]].
It is determined, up to a constant of modulus $1$, by the modulus of its boundary values.
It is the exponential of the holomorphic function whose real part is the Poisson integral of the logarithm of that boundary modulus.
Outer functions are the other half of the [[inner-outer factorization|X6ZQPAQF76]] of functions in [[Hardy spaces|PNNNFJVF6W]].

## Explanations

### L1 human

Let $\varphi$ be a positive weight on the circle whose logarithm is integrable.
There is a holomorphic function with $\abs{F} = \varphi$ on the circle that is as large as possible inside the disc.
It has no zeros, and $\log \abs{F}$ is the harmonic function with boundary values $\log \varphi$.
That function is the outer function with modulus $\varphi$.
For example, $F(z) = 2 + z$ is outer: it has no zeros in the closed disc.
By contrast $z$ is not outer; it is [[inner|NN8XH5RJYB]], and the outer function with the same boundary modulus is the constant $1$.