5.5 Bessel’s Equation 315
Using the same method as in the preceding, an infinite series can be developed
for the solutions (see Exercise 8). The solution that is bounded atr=0, in
standard form, is called themodified Bessel function of the first kind of orderμ,
designatedIμ(λr), and its series is
Iμ(λr)=(λr
2)μ∑∞m= 01
m!(μ+m)!(λr
2) 2 m
.Summary
The differential equation
d
dr(
rdR
dr)
−
μ^2
rR+λ(^2) rR= 0
is called Bessel’s equation. Its general solution is
R(r)=AJμ(λr)+BYμ(λr)
(AandBare arbitrary constants). The functionsJμandYμare called
Bessel functions of orderμof the first and second kinds, respectively.
The Bessel function of the second kind is unbounded at the origin.
EXERCISES
- Find the values of the parameterλfor which the following problem has a
nonzero solution:
1
r
d
dr(
rddrφ)
+λ^2 φ= 0 , 0 <r<a,φ(a)= 0 ,φ( 0 )bounded.- Sketch the first few eigenfunctions found in Exercise 1.
- Show that
d
drJμ(λr)=λJ
μ′(λr),
where the prime denotes differentiation with respect to the argument. - Show from the series that
d
drJ^0 (λr)=−λJ^1 (λr).