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In this paper:

https://www.staff.science.uu.nl/~beuke106/HypergeometricFunctions/COGP.pdf

Any help how to reproduce the plot in Figure:7

enter image description here

It’s the leading order of the complex function, Equation (4.3)

enter image description here

I have written as

g[psi_]= 3/(4 psi * Conjugate[psi] * Log[Abs[5 psi ]]^2 ) * (1 - ((48 * Zeta[3.0] ) / (25* Log [Abs [5 psi]]^3 )))

It’s written in the figure’s caption that $g_{\psi \bar{\psi}}$ has been plotted against $ \psi$ ( however it’s a 3D graph! ) in a certain region of arg $\psi$ .

Any help to reproduce that will be appreciated!

Edit:

My naive trail!

enter image description here

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    $\begingroup$ Holy function, Batman! .. or holy Batman function? $\endgroup$
    – Chris K
    Sep 23 '19 at 14:22
  • $\begingroup$ Have a look at ParametricPlot3D. This way, you can use polar coordiates in the first two coordinate dimensions and the function g for the third one. This would also produce theses nice mesh lines (which cannot be obtained by the PlotRegion option in Plot3D). $\endgroup$ Sep 23 '19 at 14:52
  • $\begingroup$ Does ComplexPlot3D[g[\[Psi]], {\[Psi], -0.25 - 0.25 I, 0.25 + 0.25 I}, RegionFunction -> Function[{\[Psi]}, 0 < Arg[\[Psi]] < 2 \[Pi]/5]] help? $\endgroup$
    – murray
    Sep 23 '19 at 15:30
  • $\begingroup$ @HenrikSchumacher, My problem with parametricPlot3D or Plot3D is that the function should be in two variables coordinates, and here g is a function in $\psi$ only. So plotting it against r and $\theta$ in polar coordinates maybe needs a change of coordinates which I’m not expert about, especially i have a region of $\psi$. $\endgroup$
    – Dr. phy
    Sep 23 '19 at 16:57
  • $\begingroup$ @ChrisK, never show it like that, till your joke, nice -:) Lol! $\endgroup$
    – Dr. phy
    Sep 23 '19 at 17:33
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Clear["Global`*"]

g[psi_] = 3/(4 psi*Conjugate[psi]*
      Log[Abs[5 psi]]^2)*(1 - (48*Zeta[3]/(25*Log[Abs[5 psi]]^3)));

The paper states that the plot is for 0 <= Arg[psi] < 2 Pi/5 and that the series expansions are valid for Abs[psi] > 1

ComplexExpand[
 0 <= Arg[psi] < 2 Pi/5 && Abs[psi] > 1, {psi},
 TargetFunctions -> {Re, Im}]

(* 0 <= ArcTan[Re[psi], Im[psi]] < (2 π)/5 && Sqrt[Im[psi]^2 + Re[psi]^2] > 1 *)

ParametricPlot3D[
 {psiRe, psiIm, g[psiRe + I*psiIm]},
 {psiRe, 0, 4}, {psiIm, 0, 4},
 RegionFunction -> Function[{psiRe, psiIm, g, u, v},
   0 <= ArcTan[psiRe, psiIm] < 2 Pi/5 &&
    1 < Sqrt[psiRe^2 + psiIm^2] < 4] ,
 BoxRatios -> {1, 1, 1},
 PlotRange -> All,
 PlotPoints -> 200,
 MaxRecursion -> 5,
 AxesLabel -> (Style[#, 14, Bold] & /@ {Re[ψ], Im[ψ], "g[ψ]"})]

enter image description here

The cowls do not show up.

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  • $\begingroup$ Oh thanks anyway. About the cups, I think it’s something related to the intervals or so. Of course your graph is better than my trail in the Question ‘s edit :-) $\endgroup$
    – Dr. phy
    Sep 23 '19 at 19:32

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