Page 68 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

58 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
4.2 BESSEL FUNCTIONS
A few differential equations are so widely useful in applied mathematics that they have been
named after the mathematician who first explored their theory. Such is the case with Bessel’s
2
equation. It occurs in problems involving the Laplacian ∇ u in cylindrical coordinates when
variables are separated. Bessel’s equation is a Sturm–Liouville equation of the form

2
d u du
2
2 2
ρ 2 + ρ + (λ ρ − ν )u = 0 (4.67)
dρ 2 dρ
Changing the independent variable x = λρ, the equation becomes
2
2
x u + xu + (x − ν )u = 0 (4.68)
2

4.2.1 Solutions of Bessel’s Equation
Recalling the standard forms (4.1) and (4.2) we see that it is a linear homogeneous equation
with variable coefficients and with a regular singular point at x = 0. We therefore assume a
solution of the form of a Frobenius series (4.17).

∞
j+r
u = c j x (4.69)
j=0
Upon differentiating twice and substituting into (4.68) we find

∞
2 j+r j+r+2
[( j + r − 1)( j + r) + ( j + r) − ν ]c j x + c j x = 0 (4.70)
j=0 j=0

In general ν can be any real number. We will first explore some of the properties of the solution
when ν is a nonnegative integer, 0, 1, 2, 3, ... . First note that

( j + r − 1)( j + r) + ( j + r) = ( j + r) 2 (4.71)

Shifting the exponent in the second summation and writing out the first two terms in the first

(r − n)(r + n)c 0 + (r + 1 − n)(r + 1 + n)c 1 x
∞
j
+ [(r + j − n)(r + j + n)c j + c j−2 ]x = 0 (4.72)
j=2

0
In order for the coefficient of the x term to vanish r = n or r =−n. (This is the indicial
equation.) In order for the coefficient of the x term to vanish c 1 = 0. For each term in the