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In the following application, we use the command fzero to find the zeros
of a Bessel function, and learn in the process some important facts about this
often-used special function of applied mathematics.

Application
Bessel functions are solutions to Bessel’s differential equations of order n,
given by:

2
dy dy
2
x + x + ( xn y− ) = 0 (5.19)
dx 2 dx
There are special types of Bessel functions referred to as “of the first, second,
and third kinds.” Bessel functions of integer order appear, inter alia, in the
expression of the radiation field in cylindrically shaped resonant cavities, and
in light diffraction from a circular hole. Bessel functions of half-integer indices
(see Pb. 2.26) appear in problems of spherical cavities and scattering of elec-
tromagnetic waves. Airy functions, a member of the Bessel functions family,
appear in a number of important problems of optics and quantum mechanics.
The recursion formula that relates the Bessel function of any kind of a cer-
tain order with those of the same kind of adjacent orders is

2nZ (x) = xZ (x) + xZ n+1 (x) (5.20)
n–1
n
where Z (x) is the generic designation for all kinds of Bessel functions.
n
In this application, we concern ourselves only with the Bessel function of
the first kind, usually denoted by J (x). Its MATLAB call command is
n
besselj(n,x). In the present problem, we are interested in the root struc-
ture of the Bessel function of the first kind and of zero order.
In the program that follows, we call the Bessel function from the MATLAB
library; however, we could have generated it ourselves using the techniques
of Section 4.7 because we know the ODE that it satisfies, and its value and
that of its derivative at x = 0, namely:

Jx( = 0) = 1 and Jx( ′ = 0) = 0
0 0
The problem that we want to solve is to find the zeros of J (x) and compare
0
to these exact values those obtained from the approximate expression:
π 1 31 3779
x ≈ k 4 ( − 1) + − + +… (5.21)
0, k 24 ( π k 1)− 6π 3 − 3 15π 5 − 5
4 k 4 ( 1) k 4 ( 1)
To implement this task, edit and execute the following script M-file:

for k=1:10
p(k)=4*k-1;

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