1. 15. Legendre Functions Legendre Polynomials Orthogonality
Physical Interpretation of Generating Function
Associated Legendre Equation
Spherical Harmonics
Legendre Functions of the Second Kind

2. Schrodinger eq. for a central potential
Associated Legendre eq. 
Legendre eq.
with

3. 1. Legendre Polynomials Legendre eq. 
 x = 1 & x =  are regular singular points.

4. Frobenius Series See § 8.3 & Mathematica  Indicial eq. 
series diverges for x2 1
unless terminated
Set a1 = 0  a2 j + 1 = 0
s = 0  even order :
s = 1  odd order :

5. Generating Function    

6. 
& 
highest power of x
in coeff. of tn is n. 
Coeff. of xn in Pn(x) = Coeff. of xn tn 
Coeff. of highest power of x in Pn(x)

7. Summary
Power Expansion : Ex

8. Recurrence Relations   

9. Table 15.1. Legendre Polynomials
Mathematica

10.  
&
Eliminate x Pn1 term 

11. More Recurrence Relations
Any set of functions satisfying these recurrence relations also satisfy the Legendre ODE. Ex

12. Upper & Lower Bounds for Pn (cos )

13. 
Coeff. invariant under j   j  
Coeff. invariant under j   ( j+1) 

14. For P2n , x = 1 are global max.
For P2n+1 , x = +1 is a gloabal max,
while x = 1 is a gloabal min.
Mathematica

15. Rodrigues Formula From § 12.1 : If has self-adjoint form then
Legendre eq. :
Self-adjoint form :  

16. Coefficient of xn in Pn(x)
Coefficient of xn is :  

17. 2. Orthogonality is self-adjoint [ w(x) = 1 ]  
Pn(x) are orthogonal polynomials in [ 1, 1 ].

18. Normalization 
Let   
via Rodrigues formula : Ex

19. Legendre Series Eigenfunctions of an ODE are complete
 { Pn (x) } is completeness over [1,1] .
 For any function f (x) in [1,1] : 
unique

20. Solutions to Laplace Eq. in Spherical Coordinates
General solution :
finite  l = 0,1,2, …
Solution with no azimuthal dependence ( m = 0 ) :
Solution that is finite inside & outside a boundary sphere :

21. Example 15.2.1 Earth’s Gravitational Field
Gravitational potential U in mass-free region :
Neglect azimuthal dependence :
Earth’s radius at equator
g includes rotational effect
Note:
Let 
al dimensionless
Earth is a sphere  

22. Slightly distorted Earth with axial symmetry :
CM located at origin 
See Mathematica for proof. 
Experimental data :
Data including longitudinal dependence is described by a Laplace series (§15.5).

23. Example 15.2.2 Sphere in a Uniform Field
Grounded conducting sphere (radius r0 ) in uniform applied electric field
everywhere
 For :  

24. Surface charge density :
SI units
induced dipole moment Ex
Mathematica

25. Example 15.2.3 Electrostatic Potential for a Ring of Charge
Thin, conducting ring of radius a, centered at origin & lying in x-y plane, has total charge q.
Outside the ring,
Axial symmetry  no  dependence
Mathematica
For r > a :
On z-axis, Coulomb’s law gives :

26.  
See Eg for magnetic analog.

27. 3. Physical Interpretation of Generating Function
Leading term :
(point charge) 
for r > a. 
for r < a.

28. Expansion of 1 / | r  r |
Let : 
either r or r on z-axis

29. Electric Multipoles
Electric dipole :
point dipole
Leading term :

30. (Linear) Multipoles
Let = 2l -pole potential with center of charge at z = r.
Mono ( 20 ) -pole :
Di ( 21 ) -pole :
Quadru ( 22 ) –pole :
( 2l ) –pole :
Quadrupole
Mathematica

31. Multipole Expansion
If all charges are on the z-axis & within the interval [zm , zm ] :
for r > zm
where is the (linear) 2l –pole moment.
For a discrete set of charges qi at z = ai . 

32. If one shifts the coord origin to Z.
l is independent of coord, i.e.,  Z
iff
Multipole expansion for a general (r) are done in terms of the spherical harmonics.