---
tag: M324R146XH
kind: concept
name: Zeros of Bessel functions
msc: [33C10]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel_function
---

## Summary

The zeros of a [[Bessel function of the first kind|CZCMRDWDYW]] $J_\nu$ with $\nu > -1$ are real.
Its positive zeros form an increasing sequence $j_{\nu,1} < j_{\nu,2} < \cdots$ tending to infinity.
These numbers fix the frequencies of a vibrating circular drum and the modes of heat flow in a cylinder.
They obey interlacing and spacing rules, and they are the subject of [[Bourget's hypothesis|38D9RPAXQ8]].

## Explanations

### L1 human

The graph of $J_0(x)$ for $x > 0$ looks like a damped cosine, so it crosses the axis again and again.
The crossing points are the zeros $j_{0,1} \approx 2.405$, $j_{0,2} \approx 5.520$, and so on.
Far out they are spaced by almost exactly $\pi$, like the zeros of a cosine.
A drum of radius $1$ fixed at its rim vibrates in its lowest radially symmetric mode with frequency proportional to $j_{0,1}$.