The walls of the conducting cylindrical box of Problem 3.23 are all at zero potential, except for a disc in the upper end, defined by ρ=b<a, at potential V.
(a) Using the various forms of the Green function obtained in Problem 3.23, find three expansions for the potential inside the cylinder.
(b) For each series, calculate numerically the ratio of the potential at ρ=0, z=L/2 to the potential of the disc, assuming b=L/4=a/2. Try to obtain at least two-significant-figure accuracy. Is one series less rapidly convergent than the others? Why?
(Abramowitz and Stegun have tables; Mathematica has Bessel functions, as does the software of Press et al.)