5.5 Bessel’s Equation 313
and so forth. Allc’s with odd index are zero, since they are all multiples ofc 1.
The general formula for a coefficient with even indexk= 2 mis
c 2 m= (−^1 )m
m!(μ+ 1 )(μ+ 2 )···(μ+m)(λ
2) 2 m
c 0. (4)For integral values ofμ,c 0 is chosen by convention to be
c 0 =(λ
2)μ
·^1
μ!.
Then the solution of Eq. (1) that we have found is called theBessel function of
the first kind of orderμ:
Jμ(λr)=(
λr
2)μ∑∞m= 0(− 1 )m
m!(μ+m)!(
λr
2) 2 m. (5)
This series serves us for evaluating the function and for obtaining its proper-
ties. (See the Exercises.) However,from now on, we consider the Bessel functions
of the first kind to be as well known as sines and cosines, although less familiar.
There must be a second independent solution of Bessel’s equation, which
can be found by using variation of parameters. This method yields a solution
in the form
Jμ(λr)·∫ dr
rJ^2 μ(λr). (6)
In its standard form, the second solution of Bessel’s equation is called theBessel
function of second kind of orderμand is denoted byYμ(λr).
The most important feature of the second solution is its behavior nearr=0.
Whenris very small, we can approximateJμ(λr)by the first term of its series
expansion:
Jμ(λr)∼=(λ
2)μ 1
μ!rμ, r
1.The solution Eq. (6) then can be approximated by
constant×rμ∫ dr
r^1 +^2 μ=constant×{
ln(r), ifμ=0,
r−μ, ifμ>0.In either case, it is easy to see that
∣∣
Yμ(λr)
∣∣
→∞ asr→ 0.
Both kinds of Bessel functions have an infinite number of zeros. That is,
thereisaninfinitenumberofvaluesofα(andβ) for which
Jμ(α)= 0 , Yμ(β)= 0.