---
tag: MSXT31P944
kind: definition
name: Zeros of Bessel functions
msc: [33C10]
defines: M324R146XH
uses: [CZCMRDWDYW]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel_function
---

## Definition

Let $\nu > -1$ be real and let $J_\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$.
The positive zeros of $J_\nu$ form an infinite set with no finite accumulation point.
For $k \ge 1$, the $k$-th *zero* $j_{\nu,k}$ of $J_\nu$ is the $k$-th smallest element of this set, so that
$$0 < j_{\nu,1} < j_{\nu,2} < j_{\nu,3} < \cdots, \qquad J_\nu(j_{\nu,k}) = 0.$$

## Remarks

For $\nu > -1$, every zero of $z^{-\nu} J_\nu(z)$ in $\C$ is real and simple, so the $j_{\nu,k}$ and their negatives are all of them.
Example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$, so $j_{1/2,k} = k\pi$; the check is $\sin(k\pi) = 0$ and $\sin x \ne 0$ on $(0, \pi)$.
Numerical values: $j_{0,1} \approx 2.4048$ and $j_{1,1} \approx 3.8317$.
The zeros of $J_\nu$ and $J_{\nu+1}$ interlace: $j_{\nu,1} < j_{\nu+1,1} < j_{\nu,2} < j_{\nu+1,2} < \cdots$.
For fixed $\nu$, $j_{\nu,k+1} - j_{\nu,k} \to \pi$ as $k \to \infty$, and $j_{\nu,k} = \left( k + \frac{\nu}{2} - \frac{1}{4} \right)\pi + O\left( \frac{1}{k} \right)$ (McMahon's expansion).
A non-example: $0$ is a zero of $J_\nu$ for $\nu > 0$, but it is not one of the $j_{\nu,k}$, which are positive by definition.