---
tag: WVB60RKC3Z
kind: concept
name: Lelong number
msc: [32U25, 32U05]
sources:
  - url: https://en.wikipedia.org/wiki/Lelong_number
same_as:
  wikidata: Q6520938
  wikipedia: Lelong number
---

## Summary

The Lelong number of a [[plurisubharmonic function|FF2P7ER3QT]] $u$ at a point $x$ measures how fast $u$ tends to $-\infty$ near $x$, compared with $\log \abs{z - x}$.
The same name is used for the analogous density of a closed [[positive current|BYJHP6A906]] at a point.
For $u = \log \abs{f}$ with $f$ holomorphic, the Lelong number is the order of vanishing of $f$ at $x$.
Lelong numbers are the basic local invariant of singularities in pluripotential theory.

## Explanations

### L1 human

A plurisubharmonic function may have poles where it equals $-\infty$.
The Lelong number tells how strong such a pole is, on a logarithmic scale.
The function $\log \abs{z}$ on $\C$ has Lelong number $1$ at $0$, and $\log \abs{z^3} = 3 \log \abs{z}$ has Lelong number $3$.
At a point where the function is finite, the Lelong number is $0$.