Currently, I am trying to plot out the below equation using Mathematica.

$$Z(\rho, \phi) = 50 +\frac{200}{\pi} \sum_{odd \; m}^\infty \bigg(\frac{\rho}{a} \bigg)^m \frac{\sin (m\phi) }{m}$$

Or, if one wants, I could also rewrite the above equation as the one shown below.

$$Z(\rho, \phi) = 50 +\frac{200}{\pi} \sum_{m=0}^\infty \bigg(\frac{\rho}{a} \bigg)^{2m+1} \frac{\sin ((2m+1)\phi) }{2m+1}$$

Note that the $a$ value in the equations above stands for the radius (i.e. $\rho$) specific to my given circle.

I have attempted to write the below Mathematica code to plot out the distribution shown in the equation but to no avail.

f[n_] := 50 + 200/Pi +
  Sum[ (Rho/a)^(2n+1) Sin[(2n+1)Phi]/(2n+1), {n, 0, nmax}];
 Plot[{f[nTerms, a]}, {a, 0, 2}]

Note that I made the minor substitution of $m$ to $n$ for my above code. That aside, I am quite certain that I did the Plot section very incorrectly.

If possible, I would greatly appreciate it if anybody in the community could show and help me correctly plot for the given distribution shown in the equations above. Thank you very much!


1 Answer 1


$Assumptions = a > 0 && ρ > 0;

z[ρ_, ϕ_, a_ : 1, nmax_ : 40] := 
  50 + 200/Pi + 
   Sum[(ρ/a)^(2 n + 1) Sin[(2 n + 1) ϕ]/(2 n + 1), {n, 0, nmax}];

Plot3D[z[ρ, ϕ], {ρ, 0, 1}, {ϕ, 0, Pi},
  WorkingPrecision -> 20,
  PlotPoints -> 75,
  MaxRecursion -> 5] // Quiet

enter image description here

If the number of terms is infinite

f = z[ρ, ϕ, a, Infinity] // 
   ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify

(* (1/(4 π))(800 + 
  200 π - π ArcTan[
    a - ρ Cos[ϕ], -ρ Sin[ϕ]] + π ArcTan[
    a - ρ Cos[ϕ], ρ Sin[ϕ]] - π ArcTan[
    a + ρ Cos[ϕ], -ρ Sin[ϕ]] + π ArcTan[
    a + ρ Cos[ϕ], ρ Sin[ϕ]]) *)

   f /. a -> 1], {ρ, 0, 1}, {ϕ, 0, Pi},
  WorkingPrecision -> 20,
  PlotPoints -> 75,
  MaxRecursion -> 5]

enter image description here

  • $\begingroup$ Thank you so much for the help! I did try copying your line of code onto the Mathematica on my device to see how it worked out; however, the plot refused to show up - only allowing an empty but interactive box to show on the screen. I did try to retype the entire code while writing ϕ as \[CurlyPhi] and ρ as "esc" rho "esc". Unfortunately, the problem persists. Do you perhaps have any idea why this "empty plot" problem exists? Is it perhaps due to the fact that I only have a "basic" plan on Mathematica? $\endgroup$
    – Athenian
    Sep 30, 2020 at 19:26
  • 1
    $\begingroup$ Try restarting Mathematica then copy and paste the code into a new notebook. $\endgroup$
    – Bob Hanlon
    Sep 30, 2020 at 19:29
  • $\begingroup$ Thank you for the suggestion! While it did take a couple more attempts, I was able to get it working once more. Beyond that, thank you for your comprehensive post! $\endgroup$
    – Athenian
    Sep 30, 2020 at 19:40

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