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

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

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

4. (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

5. 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 . 

6. 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.

7. 4. Associated Legendre Equation
Let  
Set 
Mathematica

8. Frobenius Series 
with indicial eqs.
By definition,  or
Mathematica

9. Series diverges at x = 1 unless terminated.
For s = 0 & a1= 0 (even series) :
( l,m both even or both odd ) 
Mathematica
For s = 1 & a1=0 (odd series) :
( l,m one even & one odd ) 
Plm = Associated Legendre function

10. Relation to the Legendre Functions
Generalized Leibniz’s rule :   

11. Associated Legendre function :
Set
Associated Legendre function :
()m is called the Condon-Shortley phase. Including it in Plm means Ylm has it too. 
Rodrigues formula :   
Mathematica

12. Generating Function & Recurrence   
( Redundant since Plm is defined only for l  |m| 0. )
&

13.  
as before    

14. 
( Redundant since Plm is defined only for l  |m| 0. )
& 

15. Recurrence Relations for Plm
(1) = (15.88)
(2)
(1) :
(3)
(3)  (2) :
(15.89)

16. Table 15.3 Associated Legendre Functions
Using
one can generate all Plm (x) s from the Pl (x) s.
Mathematica

17. Example 15.4.1. Recurrence Starting from Pmm (x)
no negative powers of (x1)      

18. l = m 
l = m+k1 
E.g., m = 2 :

19. Parity & Special Values
Rodrigues formula : 
Parity
Special Values : Ex

20. Orthogonality
Plm is the eigenfunction for eigenvalue of the Sturm-Liouville problem
where
Lm is hermitian  ( w = 1 )
Alternatively : 

21. No negative powers allowed
For p  q , let 
&
only j = q ( x = + 1) or j = kq ( x =  1 ) terms can survive

22. p  q :
For j > m : 
For j < m + 1 : 

23. p  q :  
Only j = 2m term survives  

24. Ex
B(p,q)  

25.  For fixed m, polynomials { Ppm (x) } are orthogonal with weight ( 1  x2 )m .
Similarly

26. Example 15.4.2. Current Loop – Magnetic Dipole
Biot-Savart law (for A , SI units) :
By symmetry :
Outside loop : 
E.g
Mathematica  

27. 
For r > a :  

28. 
For r > a :

29. on z-axis : or  
(odd in z)

30. Biot-Savart law (SI units) :
Cartesian coord:  

31. 
For r > a :

32. s 1 2 s
3/2
15/8 s 1 2 s 1
3/4
5/8

33. Electric dipole :  