---
tag: 9P87VQZ0P7
kind: concept
name: Anger function
msc: [33C10]
sources:
  - url: https://en.wikipedia.org/wiki/Anger_function
same_as:
  wikidata: Q3754606
  wikipedia: Anger function
---

## Summary

The Anger function $\mathbf{J}_\nu$ is the function obtained by using Bessel's integral formula for an arbitrary order $\nu$.
For integer $\nu$ it equals the [[Bessel function of the first kind|CZCMRDWDYW]] $J_\nu$; for other $\nu$ it does not.
Its companion is the Weber function $\mathbf{E}_\nu$, defined with a sine in place of the cosine.
Both solve inhomogeneous forms of Bessel's equation.

## Explanations

### L1 human

Bessel found that $J_n(z)$ is the average of $\cos(n\tau - z\sin\tau)$ over $0 \le \tau \le \pi$ when $n$ is a whole number.
One can compute the same average when $n$ is replaced by any number $\nu$.
The result is the Anger function.
When $\nu$ is not a whole number, it is no longer a Bessel function, which shows that Bessel's integral is special to integer orders.