1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.10 Some Applications of Legendre Polynomials 349

Again we see that the ratio containingmust be constant, say,−μ^2 .Hence,
we have two separate problems for the functionsandR:
(
sin(φ)′

)′

+μ^2 sin(φ)= 0 , 0 <φ<π,

(φ)bounded atφ= 0 ,π,

(

ρ^2 R′

)′

−μ^2 R+λ^2 ρ^2 R= 0 , 0 <ρ<a,

R(a)= 0 ,

R(ρ)bounded at 0.

The first of these problems is now quite familiar, and we know its solution to
be

μ^2 n=n(n+ 1 ), n(φ)=Pn

(

cos(φ)

)

, n= 0 , 1 , 2 ,....
The second problem is less familiar. In standard form, the differential equa-
tion is

R′′+^2

ρ

R′−μ

2

ρ^2

R+λ^2 R= 0.

Comparison with the four-parameter form of Bessel’s equation (Eq. (1) of Sec-
tion 5.8) showsα=− 1 /2,γ=1, andp^2 =μ^2 +α^2 .Sinceμ=n(n+ 1 ),
p^2 =n^2 +n+^14 ,andthenp=n+^12. Thus, the general solution of the differ-
ential equation is

Rn(ρ)=ρ−^1 /^2

[

AJn+ 1 / 2 (λρ)+BYn+ 1 / 2 (λρ)

]

.

The fact that the Bessel functions of the second kind,Yp(λρ),areun-
bounded atρ=0 allows us to discard them from the solution, leaving

Rn(ρ)=ρ−^1 /^2 Jn+ 1 / 2 (λρ)

as the bounded solution. These functions occur frequently in problems in
spherical coordinates. Sometimes the functions

jn(z)=

√

π

2 zJn+^1 /^2 (z),

calledspherical Bessel functions of the first kind of order n,areintroduced.As
noted in Section 5.8, there is a relation to sines and cosines:

j 0 (z)=sin(z)/z,

j 1 (z)=

(

sin(z)−zcos(z)

)

/z^2 ,

j 2 (z)=

(

( 3 −z^2 )sin(z)− 3 zcos(z)

)

/z^3.
We have yet to satisfy the boundary conditionRn(a)=0. This cannot be
done by formula, except forn=0. In this case,R 0 (a)=0comesdownto