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15.3 Bessel Functions 555

TABL E 15.1 Some Zeros of Bessel functions

J n j 1 j 2 j 3 j 4 j 5
2.405 5.520 8.654 11.792 14.931
J 0
3.832 7.016 10.173 13.323 16.470
J 1
5.135 8.417 11.620 14.796 17.960
J 2
6.379 9.760 13.017 16.224 19.410
J 3
7.586 11.064 14.373 17.616 20.827
J 4

EXAMPLE 15.7
We will compute some terms in the Fourier-Bessel expansion of f (x) = x(1 − x) on [0,1], with
ν = 1. In this case the eigenfunctions are J 1 ( j n x) and the j n ’s are the positive zeros of J 1 .The
coefficients are
1 2
2 x (1 − x)J 1 ( j n x)dx
c n = 0
2
J ( j n )
2
and, because x(1 − x) is twice differentiable on [0,1], we will have
∞

x(1 − x) = c n J ν ( j n x)dx
n=1
for 0 < x < 1. We will compute the first four coefficients to illustrate the ideas involved. First we
need j 1 ,··· , j 4 . These can be obtained from tables, or from MAPLE by the command

evalf(BesselJZeros(ν,n));

with ν = 1 in this example and n successively chosen to be 1,2,3,4. The output is
j 1 ≈ 3.83170597, j 2 ≈ 7.01558667, j 3 ≈ 10.17346814, j 4 ≈ 13.32369194.
Using this value for j 1 , c 1 is approximately

2 1
2
c 1 ≈ x (1 − x)J 1 (3.83170597x)dx.
J 2 (3.83170597) 2 0
To carry out this computation, first approximate the denominator J ( j 1 ), using a software
2
2
package. If MAPLE is used, the denominator is the square of the number computed as
evalf (BesselJ(2,3.83170597));

The integral in the numerator is computed as

evalf int((x ∧ 2) * (1-x) * BesselJ(1,x)(3.83170597 * x),x=0..1);

This computation yields

c 1 ≈ 0.45221702.
By repeating this calculation for (n =2,3,4) in turn with the appropriate j n inserted, we similarly
approximate

c 2 ≈−0.03151859,c 3 ≈ 0.03201789,c 4 ≈−0.00768864.

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