We discussed the solution of Laplace's equation with the separation of variables. We focused on spherical coordinates, and discussed the meaning of the various constants appearing in the differential equation written in spherical coordinates. In particular, we first discussed the behavior of both the azymuthal and polar angular dependences of the Laplace equation, by making a connection with our example on Scattering Theory (see also previous pages).
The question of multivalued complex functions came up. We discussed those in relation to solutions of the azymuthal angular part that can be written in the form: exp(-i m &phi) with "m" integer.
We attacked the problem of the dielectric sphere in a uniform electrostatic field, and discussed the properties of the Legendre polynomials. We covered Chapter 3 , and Sections 4.1-4.2 of Jackson.

1) Study the function 1/&radic z. Find its real and imaginary parts. How many branches does this functions have? How many Riemann sheets?

2) Find the solutions to z⁴ =16, and plot them on the complex plane.

3) Use the recursion relations for the Legendre Polynomials listed in your textbook to prove that the derivatives, P_l'(x) are orthogonal in the interval [-1,1], with weighting function (1-x²) Due by the end of the week!