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15.3 Bessel Functions 533

and 7. Let n be a nonnegative integer. Prove that
(2n)!
R = (x − x 0 ) + (y − y 0 ) + (z − z 0 ) . P 2n+1 (0) = 0and P 2n (0) = (−1) n .
2
2
2
2 (n!) 2
2n
8. Expand each of the following polynomials in a series
(a) Use the law of cosines to write
of Legendre polynomials.
1 (a) 1 + 2x − x 2
.
ϕ(x, y, z) = 2 3
d 1 − 2(r/d)cos(θ) + (r/d) 2 (b) 2x + x − 5x
(c) 2 − x + 4x 4
2
(b) If r < d, use the generating function for Legendre
polynomials to show that In each of Problems 9 through 14, find the first five coef-
ficients in the Fourier-Legendre expansion of f (x) on
∞ [−1,1]. Graph the function and the partial sum of these
1

n
ϕ(r) = P n (cos(θ))r . first five terms on the same set of axes, for −3≤ x ≤3. The
d n+1
n=0 expansion only represents the function on (−1,1), but it is
instructive to see how the partial sums and the function are
(c) If r > d, use the generating function to show that
unrelated outside this interval.
∞
1 n −n 9. f (x) = sin(πx/2)
ϕ(r) = d P n (cos(θ))r .
r 10. f (x) = e −x
n=0
2
11. f (x) = sin (x)
6. Show that 12. f (x) = cos(x) − sin(x)

∞
−1for −1 ≤ x ≤ 0
1 1
P n (1/2) = √ . 13. f (x) =
2 n+1 3 1 for 0 < x ≤ 1.
n=0
14. f (x) = (x + 1)cos(x)
15.3 Bessel Functions
This section is devoted to Bessel functions, Bessel’s differential equation, Fourier-Bessel
eigenfunction expansions and some applications.
Friedrich Wilhelm Bessel (1784 - 1846) was a mathematician and director of the astro-
nomical observatory in Königsberg. He obtained series which would later be known as Bessel
functions in solving a problem known as Kepler’s problem. The same functions had appeared
in the 1730’s in work by the Swiss natural philosopher Daniel Bernoulli, who was attempt-
ing to describe oscillations in a suspended heavy chain. Joseph Fourier also encountered these
functions in his treatise on heat diffusion, which carried the seeds of modern day Fourier
analysis. We will discuss some of these applications as we develop the functions named for
Bessel.
To do this we will need the gamma function, which is used in writing Bessel functions.

15.3.1 The Gamma Function

The gamma function is defined by
∞

e dt.

(x) = t x−1 −t
0

This integral converges for x > 0. We will use the following property of
(x).

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October 14, 2010 15:20 THM/NEIL Page-533 27410_15_ch15_p505-562