---
tag: RHDBK62123
kind: concept
name: Nontangential limit
msc: [31B25, 31A20, 30H05]
sources:
  - url: https://en.wikipedia.org/wiki/Fatou%27s_theorem
---

## Summary

A function on the unit ball has a nontangential limit at a boundary point if it converges along every approach inside a cone.
The cones have their vertex at the boundary point.
Such cones, called Stolz angles in the plane, keep the approach away from tangential directions.
Nontangential limits are the right notion of boundary value for harmonic and holomorphic functions.
[[Fatou's theorem|Q2N917JQ6A]] says that Poisson integrals and bounded holomorphic functions have them almost everywhere.

## Explanations

### L1 human

Walk towards a point on the boundary circle of the unit disc.
If you come in at an angle, staying inside a fixed wedge with tip at the point, you approach nontangentially.
A function has a nontangential limit if it tends to one value along all such walks.
Walks that hug the circle, tangential walks, are allowed to give other values.
For example, $\exp\left(-\frac{1 + z}{1 - z}\right)$ tends to $0$ at $1$ nontangentially, but has modulus $1/e$ on a circle through $1$ tangent to the unit circle.