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I have looked around for a code that can help to plot the magnitude and phase of the following sum

Sum[(BesselJ[n, r] e^(I n ϕ))/I^n, {n, -N, N}]

But I am not able to find an example. I would like to plot this for various levels of n. Any help very appreciated!

Thanks

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  • $\begingroup$ Sum[BesselJ[n, r] Sin[I* n *\[Phi]]/I^n, {n, -k, k}] performs 0. $\endgroup$
    – user64494
    Commented Apr 14, 2021 at 14:03
  • $\begingroup$ N has a built-in meaning and should not be used as a user-defined variable/constant. Sin[I n \[Phi]] is the proper syntax and spelling; however, the sum is zero so the plot won't be very interesting. $\endgroup$
    – Bob Hanlon
    Commented Apr 14, 2021 at 14:04
  • $\begingroup$ @BobHanlon See changes in the formula $\endgroup$
    – user287546
    Commented Apr 14, 2021 at 15:11
  • $\begingroup$ e should be E (all built - in names start with a capital letter) . Use Plot3D to plot against both r and phi for a given number of terms. $\endgroup$
    – Bob Hanlon
    Commented Apr 14, 2021 at 15:46
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    $\begingroup$ Use the documentation. Plot3D and Pi are misspelled. There are syntax errors. You cannot plot a complex function with Plot3D, use Abs or Arg of the function to obtain a real-valued function. The range of r and phi appear to be interchanged. $\endgroup$
    – Bob Hanlon
    Commented Apr 14, 2021 at 19:03

1 Answer 1

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As already been advised in the comments:

  1. Don't use N - it is reserved.
  2. Use E instead of e. e is just a parameter.

There are 2 functions that can help you: ComplexPlot,ComplexPlot3D. You can read about them here. I will provide you an example:

nf = 4; (*Final n*)
f1[r_, \[Phi]_] :=
  Sum[(BesselJ[n, r]*E^(I n \[Phi]))/I^n, {n, -nf, nf}];
(*Changing to singal variable representation *)
f2[z_] := f1[Abs[z], Arg[z]];
ComplexPlot[f2[z], {z, -1 - I, 1 + I}] 
ComplexPlot3D[f2[z], {z, -1 - I, 1 + I}]

The output: Color -Phase Color - Phase, Height - Manititude

You can also read about the Manipulate option to write a code where you change nf with a dynamic slide. Generally, if you want to show several graphics combined you would use the Show function, but as you use the color of the graph to represent the phase, I would not advise doing so. Plot for every n separately.

Hope that helps

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  • $\begingroup$ Many thanks, this is exactly what I was looking for. $\endgroup$
    – user287546
    Commented Apr 15, 2021 at 14:34
  • $\begingroup$ I tried your suggestion with Manipulate. Does the command fit in at the end? (Final n) u[r_, [Phi]_] := Sum[(BesselJ[n, r]*E^(I n [Phi])) I^n, {n, -nf, nf}]; (Changing to single variable representation) u1[z_] := u[Abs[z], Arg[z]]; ComplexPlot[u1[z], {z, -1 - I, 1 + I}] Manipulate[u1[z], {nf, {1, 2, 3, 5}}] ? $\endgroup$
    – user287546
    Commented Apr 15, 2021 at 14:52
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    $\begingroup$ @user287546 You will have to use Plot inside manipulate: Manipulate[Plot[u1[z],{z,0,1}],{nf,1,2,3,5}]. Please read more about Manipulate. $\endgroup$
    – Ymir
    Commented Apr 15, 2021 at 16:24
  • $\begingroup$ Ok, thank you for this $\endgroup$
    – user287546
    Commented Apr 15, 2021 at 17:46
  • $\begingroup$ Hi @Ymir, I tried to extent your code to a certain defined domain of phi and r. But this works only for complexplot and not for complexplot3D. Are there limitations of Complexplot3d for this case? nf = 5;(Final n)[u[r_, [Phi]_] := Sum[(BesselJ[n, r]*E^(I n [Phi])) I^n, {n, -nf, nf}], {r, 0, 4}, {[Phi], 0, 2 [Pi]}]; (Changing to single variable representation) u1[z_] := u[Abs[z], Arg[z]]; ComplexRegionPlot[ Abs[u1[z]] <= 1 && Abs[Arg[u1[z]]] > [Pi]/4, {z, -1 - I, 1 + I}, PlotStyle -> Orange] $\endgroup$
    – user287546
    Commented Apr 16, 2021 at 13:17

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