1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.9 Spherical Coordinates; Legendre Polynomials 343

Figure 14 The nodal curves of the zonal harmonics are the parallels
(φ=constant) on a sphere, wherePn(cos(φ))=0. The nodal curves are shown
in projection forn= 1 , 2 , 3 ,4. See the CD for color versions.

Summary

The solution of the eigenvalue problem

(

( 1 −x^2 )y′

)′

+μ^2 y= 0 , − 1 <x< 1 ,

y(x)bounded atx=−1andatx= 1 ,

isy(x)=Pn(x),μ^2 n=n(n+ 1 ),n= 0 , 1 , 2 ,....

The solution of the eigenvalue problem

(

sin(φ)′

)′

+μ^2 sin(φ)= 0 ,
(φ)bounded atφ=0andatφ=π,
is(φ)=Pn(cos(φ)),μ^2 n=n(n+ 1 ),n= 0 , 1 , 2 ,....
TheLegendrepolynomialsPn(cos(φ))are often calledzonal harmonicsbe-
cause their nodal lines (loci of solutions ofPn(cos(φ))=0) divide a sphere
into zones, as shown in Fig. 14.

EXERCISES

1. Equation (4) may be solved by assuming

(φ)=^1

2

a 0 +

∑∞

1

akcos(kφ).

Find the relations among the coefficientsakby computing the terms of the

equation in the form of series. Use the identities

sin(φ)sin(kφ)=

1

2

[

cos

(

(k− 1 )φ

)

−cos

(

(k+ 1 )φ

)]

,

sin(φ)cos(kφ)=−^1

2

[

sin

(

(k− 1 )φ

)

+sin

(

(k+ 1 )φ

)]

.