# Roots of a Bessel function

Nick Trefethen, September 2010

(Chebfun example roots/BesselRoots.m)

Here is the Bessel function J0 on the interval [0,100].

J0 = chebfun(@(x) besselj(0,x),[0 100]); figure, plot(J0,'linewidth',1.6), grid on title('Bessel function J_0','fontsize',16)

We can find its roots like this:

r = roots(J0); hold on, plot(r,J0(r),'.r','markersize',20)

The number of roots can be found with the LENGTH command:

number_of_roots = length(r)

number_of_roots = 32

Suppose you wanted to know the numbers of roots in various intervals [a,b]. You could define an anonymous function:

rootsab = @(a,b) length(roots(chebfun(@(x) besselj(0,x),[a b])));

For example:

```
tic
disp('Number of roots between 1000000 and 1001000:')
n = rootsab(1000000,1001000)
toc
```

Number of roots between 1000000 and 1001000: n = 318 Elapsed time is 0.133096 seconds.