Physics 4392
Final Exam
Griffiths: Problem 3.39
Prof. Olness

You need to plot this. I will show you how. you need to make your own function using your answer to the problem.

<<Graphics`ParametricPlot3D`

?CylindricalPlot3D

inside  = Sum[f[n], {n, 1, 5, 2}]

inside =Sum[YOUR_FUNCTION  _HERE ,{n, 1, Infinity, 2}]

(* 
      Check that the result is purely numerical  and real *) 
inside   // 
        Re  /. {r -> 2, ϕ -> 3}  // N

p1 = 
      CylindricalPlot3D[ inside  // Re , {r, 0, 1}, {ϕ, 0, 2  π}, 
        BoxRatios -> {1, 1, 1}, PlotPoints -> 15]

outside =Sum[ YOUR_FUNCTION  _HERE ,{n, 1, Infinity, 2}]

{inside, outside } /. {ϕ -> {0, π}}

{inside == outside } /. {r -> 1}

p2 = 
      CylindricalPlot3D[ 
        inside  // Re , {r, 1, 4}, {ϕ, -0.001, 2  π}, 
        BoxRatios -> {1, 1, 1}, PlotPoints -> 15]

Show[p1, p2]

Sample plot: (not the real answer)

CylindricalPlot3D[ 
      r^2 Cos[4  ϕ] , {r, 0, 2}, {ϕ, 0, 2  π}, 
      BoxRatios -> {1, 1, 1}, PlotPoints -> 50]