Read Chapter 3 of Jackson.

1. Convince yourself that Eqs. 3.62 and 3.70 are correct by expanding the expressions to second order. That is, verify the following:

   Pl( ^ r   · ^ r   ¢   ) = 4 p 2 l + 1 l å m = -l  Y*lm( ^ r   )Ylm( ^ r   ¢   )

   (1)

   and

   1 |r - r¢| = ¥ å l = 0  l å m = -l  4 p 2l+1 rl < rl+1 >   Y*lm( ^ r   )Ylm( ^ r   ¢   ).

   (2)

2. Consider a charge distribution of the form:

   r(r, q, f) = r0 r2 e-ar cos2(q),

   (3)

   where r₀ and a are constants.

   1. Express r(r, q, f) as a sum of radial functions time spherical harmonic functions in the form:

      r(r, q, f) = å lm  rlm(r)   Ylm(q, f).

      (4)

   2. Using the results of the previous problem, find the corresponding electrostatic potential which vanishes at ¥.