# Plotting a function in 3 dimensions within a domain

In this paper:

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

Any help how to reproduce the plot in Figure:7

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

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!

• Holy function, Batman! .. or holy Batman function? Sep 23 '19 at 14:22
• 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). Sep 23 '19 at 14:52
• 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? Sep 23 '19 at 15:30
• @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 $\psi$. Sep 23 '19 at 16:57
• @ChrisK, never show it like that, till your joke, nice -:) Lol! Sep 23 '19 at 17:33

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]"})]
`

The cowls do not show up.

• 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 :-) Sep 23 '19 at 19:32