# ContourPlot3D doesn't even complete after days of running [closed]

I'm using the code

ContourPlot3D[
NumericQ[
Integrate[
Piecewise[
{{Exp[-1/(1 - x^4 - y^4)]*
Exp[I*ω*10*({Cos[ϕ], Sin[θ]} - {Cos[π/2], Sin[π/2]})],
Sqrt[x^2 + y^2] < 1},
{0, Sqrt[x^2 + y^2] >= 1}},
PerformanceGoal -> "Speed"],
{x, -2, 2}, {y, -2, 2}]],
{ϕ, 0, 2*π}, {θ, 0, 2*π}, {ω, -2, 2},
PerformanceGoal -> "Speed", MaxRecursion -> 0]


I had the parameters set higher before, namely {x,-10,10}, etc., and I didn't use the MaxRecursion nor PerformanceGoal, but after four days of running I still had no result. Even now, I have had it running for most of the day and still no result.

Is this normal? Can I speed it up?

Edit: Note that the piecewise function I'm trying to plot is:

$$\frac{e^{10i\omega}}{40\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\left(-\frac{1}{1-x^{4}-y^{4}}\right)\exp\left(10i\omega\left(\begin{pmatrix} \cos\phi\\\sin\theta \end{pmatrix}-\begin{pmatrix} \cos\frac{\pi}{2}\\\sin\frac{\pi}{2} \end{pmatrix}\right)\begin{pmatrix} x\\y \end{pmatrix}\right)\,dx\,dy, \text{ if }\sqrt{x^2+y^2}<1\text{ and }0\text{ otherwise}.$$

## closed as off-topic by m_goldberg, corey979, Sascha, Feyre, MarcoBDec 26 '16 at 22:33

• The question does not concern the technical computing software Mathematica by Wolfram Research. Please see the help center to find out about the topics that can be asked here.
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• The code makes no sense. There are many braces that make no sense where they are. There is a PerformanceGoal -> "Speed" inside Piecewise. NumericQ doesn't seem to make much sense here either. – AccidentalFourierTransform Dec 25 '16 at 20:01
• @AccidentalFourierTransform Indeed, the PeformanceGoal->"Speed" was misplaced in Piecewise. And NumericQ is there otherwise Mathematica does not perform the integral operation since it has non-numerical values. – Jason Born Dec 25 '16 at 20:04
• the integrand has non-numerical values because your Piecewise function is not correctly built. Read the documentation on how this function is used. – AccidentalFourierTransform Dec 25 '16 at 20:06
• @AccidentalFourierTransform I already read it and I just made the code a bit more efficient by changing it to Piecewise[{{Exp[-1/(1 - x^4 - y^4)]* Exp[I*\[Omega]*10*({Cos[\[Phi]], Sin[\[Theta]]} - {Cos[\[Pi]/2], Sin[\[Pi]/2]})], Sqrt[x^2 + y^2] < 1}}, 0]. I think the problem maybe lies with the two dimensional vectors {Cos[\[Phi]], Sin[\[Theta]]} and the other one? – Jason Born Dec 25 '16 at 20:27
• I'm voting to close this question as off-topic because it is too localized; i.e, it applies only to the local situation and needs of its poster and answers will not benefit others. – m_goldberg Dec 26 '16 at 16:42

Your problem has very little to do ContourPlot3D. The problem lies in the Piecewise expression you are giving NIntegrate. It simply doesn't evaluate to a scalar for a large part of its domain. Consider

With[{ϕ = 1, θ = 1, ω = 1},
With[{x = -.5, y = .5},
Piecewise[
{{Exp[-1/(1 - x^4 - y^4)] Exp[I ω 10 ({Cos[ϕ], Sin[θ]} - {0, 1})],
Sqrt[x^2 + y^2] < 1},
{0, Sqrt[x^2 + y^2] >= 1}}]]]


This returns

{0.203152 - 0.245827 I, -0.00462201 - 0.318873 I}

A list of complex numbers is simply not going to work as the value of an integrand. NIntegrate needs a integrand that evaluates to real or complex scalar at each point in its domain. For what you want to do, it had better be a real.

You need to reformulate the expression you are trying integrate.

• When I formulate it like ContourPlot3D[ Integrate[ Piecewise[{{Exp[-1/(1 - x^4 - y^4)]* Exp[I*\[Omega]*10*(x*Cos[\[Phi]] + y*Sin[\[Theta]] - x*Cos[\[Pi]/2] - y*Sin[\[Pi]/2])], Sqrt[x^2 + y^2] < 1}}, 0], {x, -10, 10}, {y, -10, 10}], {\[Phi], 0, 2*\[Pi]}, {\[Theta], 0, 2*\[Pi]}, {\[Omega], -10, 10}] I get the error SystemException["MemoryAllocationFailure"]. Likewise when I use @Re before the piecewise expression. – Jason Born Dec 26 '16 at 13:06