# Integral taking forever to compute

I have this complex function $$wig2$$ which I am trying to double integrate with respect to 2 variables $$q$$ and $$p$$. Using the code below, I have tried to run it several times; however, it takes forever to compute and doesn't produce a final answer. It just keeps on running. I have tried leaving it for hours but it didn't work.

I have also tried using NIntegrate by fixing limits of $$q$$ from $$-5$$ to $$5$$ but it says the integrand has evaluated to a non-numerical value at the sampling points.

Please guide me why this is happening and how can I fix this?

wig2 = (1/(81*Pi))*4*Abs[(97 - 648*I*Sqrt[2]*p - 2992*p^2 + 2624*I*Sqrt[2]*p^3 + 1088*p^4 - 648*Sqrt[2]*q + 5984*I*p*q + 7872*Sqrt[2]*p^2*q - 4352*I*p^3*q + 2992*q^2 - 7872*I*Sqrt[2]*p*q^2 - 6528*p^2*q^2 - 2624*Sqrt[2]*q^3 + 4352*I*p*q^3 + 1088*q^4 + 64*(17 + 64*p^4 + 64*I*p^3*(3*Sqrt[2] - 4*q) - 88*Sqrt[2]*q + 304*q^2 - 192*Sqrt[2]*q^3 + 64*q^4 + 16*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*I*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3))*Conjugate[p]^4 - 8*(81*Sqrt[2] + 704*Sqrt[2]*p^4 - 1000*q + 2144*Sqrt[2]*q^2 - 3520*q^3 + 704*Sqrt[2]*q^4 - 704*I*p^3*(-5 + 4*Sqrt[2]*q) - 32*p^2*(67*Sqrt[2] - 330*q + 132*Sqrt[2]*q^2) + 8*I*p*(-125 + 536*Sqrt[2]*q - 1320*q^2 + 352*Sqrt[2]*q^3))*Conjugate[q] + 16*(187 + 1216*p^4 + 128*I*p^3*(25*Sqrt[2] - 38*q) - 1072*Sqrt[2]*q + 4240*q^2 - 3200*Sqrt[2]*q^3 + 1216*q^4 + 16*p^2*(-265 + 600*Sqrt[2]*q - 456*q^2) - 16*I*p*(67*Sqrt[2] - 530*q + 600*Sqrt[2]*q^2 - 304*q^3))*Conjugate[q]^2 - 64*(41*Sqrt[2] + 192*Sqrt[2]*p^4 - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4 - 64*I*p^3*(-17 + 12*Sqrt[2]*q) - 32*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) + 8*I*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3))*Conjugate[q]^3 + 64*(17 + 64*p^4 + 64*I*p^3*(3*Sqrt[2] - 4*q) - 88*Sqrt[2]*q + 304*q^2 - 192*Sqrt[2]*q^3 + 64*q^4 + 16*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*I*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3))*Conjugate[q]^4 + 64*Conjugate[p]^3*(-192*I*Sqrt[2]*p^4 - 64*p^3*(-17 + 12*Sqrt[2]*q) + 32*I*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) + 8*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3) - I*(41*Sqrt[2] - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4) - 4*(-64*I*p^4 + 64*p^3*(3*Sqrt[2] - 4*q) - 16*I*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3) + I*(-17 + 88*Sqrt[2]*q - 304*q^2 + 192*Sqrt[2]*q^3 - 64*q^4))*Conjugate[q]) + 16*Conjugate[p]^2*(-187 - 1216*p^4 - 128*I*p^3*(25*Sqrt[2] - 38*q) + 1072*Sqrt[2]*q - 4240*q^2 + 3200*Sqrt[2]*q^3 - 1216*q^4 + p^2*(4240 - 9600*Sqrt[2]*q + 7296*q^2) + 16*I*p*(67*Sqrt[2] - 530*q + 600*Sqrt[2]*q^2 - 304*q^3) + 12*(41*Sqrt[2] + 192*Sqrt[2]*p^4 - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4 - 64*I*p^3*(-17 + 12*Sqrt[2]*q) - 32*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) + 8*I*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3))*Conjugate[q] + 24*(-17 - 64*p^4 - 64*I*p^3*(3*Sqrt[2] - 4*q) + 88*Sqrt[2]*q - 304*q^2 + 192*Sqrt[2]*q^3 - 64*q^4 + p^2*(304 - 576*Sqrt[2]*q + 384*q^2) + 8*I*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3))*Conjugate[q]^2) + 8*Conjugate[p]*(704*I*Sqrt[2]*p^4 + 704*p^3*(-5 + 4*Sqrt[2]*q) - 32*I*p^2*(67*Sqrt[2] - 330*q + 132*Sqrt[2]*q^2) - 8*p*(-125 + 536*Sqrt[2]*q - 1320*q^2 + 352*Sqrt[2]*q^3) + I*(81*Sqrt[2] - 1000*q + 2144*Sqrt[2]*q^2 - 3520*q^3 + 704*Sqrt[2]*q^4) + 4*(-1216*I*p^4 + 128*p^3*(25*Sqrt[2] - 38*q) - 16*I*p^2*(-265 + 600*Sqrt[2]*q - 456*q^2) - 16*p*(67*Sqrt[2] - 530*q + 600*Sqrt[2]*q^2 - 304*q^3) + I*(-187 + 1072*Sqrt[2]*q - 4240*q^2 + 3200*Sqrt[2]*q^3 - 1216*q^4))*Conjugate[q] + 24*(192*I*Sqrt[2]*p^4 + 64*p^3*(-17 + 12*Sqrt[2]*q) - 32*I*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) - 8*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3) + I*(41*Sqrt[2] - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4))*Conjugate[q]^2 + 32*(-64*I*p^4 + 64*p^3*(3*Sqrt[2] - 4*q) - 16*I*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3) + I*(-17 + 88*Sqrt[2]*q - 304*q^2 + 192*Sqrt[2]*q^3 - 64*q^4))*Conjugate[q]^3))/E^(2*Abs[-(1/Sqrt[2]) + I*p + q]^2)/ ((Sqrt[2] - 4*I*p - 4*q)^2*(Sqrt[2] + 4*I*Conjugate[p] - 4*Conjugate[q])^2)];

abs = ComplexExpand[Abs[wig2]]];
intofwig = Integrate[abs, q];
doubint = Integrate[intofwig, p];
delta = Simplify[doubint] - 1;


• You write Integrate[abs, q]; but where is abs defined? May be you forgot the code for that? Commented Aug 20, 2023 at 8:09
• Better is use: Conjugate[q] == Abs[q]^2/q and Conjugate[p] == Abs[p]^2/p for analytical solution if closed-form exist !. I Try with: NIntegrate[wig2, {q, -5, 5}, {p, -5, 5}, Method -> "LocalAdaptive"] and give me: 0.6804 with no errors. Commented Aug 20, 2023 at 8:25
• @Nasser I edited it now, I wrote it in the code but there was problem with it displaying here Commented Aug 20, 2023 at 8:35
• NIntegrate[abs, {q, -5, 5}, {p, -5, 5}, Method -> "LocalAdaptive"] Commented Aug 20, 2023 at 9:12
• The wig2 expression is to complicated to computed analyticaly. Probably only way is NIntegrate . Commented Aug 20, 2023 at 9:38

By going to polar coordinates the integrand becomes a lot simpler and can potentially be studied in more detail.

First, define wig2 without the Abs function:

wig2[p_, q_] = (1/(81*Pi))*4*((97 - 648*I*Sqrt[2]*p - 2992*p^2 + 2624*I*Sqrt[2]*p^3 + 1088*p^4 - 648*Sqrt[2]*q + 5984*I*p*q + 7872*Sqrt[2]*p^2*q - 4352*I*p^3*q + 2992*q^2 - 7872*I*Sqrt[2]*p*q^2 - 6528*p^2*q^2 - 2624*Sqrt[2]*q^3 + 4352*I*p*q^3 + 1088*q^4 + 64*(17 + 64*p^4 + 64*I*p^3*(3*Sqrt[2] - 4*q) - 88*Sqrt[2]*q + 304*q^2 - 192*Sqrt[2]*q^3 + 64*q^4 + 16*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*I*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3))*Conjugate[p]^4 - 8*(81*Sqrt[2] + 704*Sqrt[2]*p^4 - 1000*q + 2144*Sqrt[2]*q^2 - 3520*q^3 + 704*Sqrt[2]*q^4 - 704*I*p^3*(-5 + 4*Sqrt[2]*q) - 32*p^2*(67*Sqrt[2] - 330*q + 132*Sqrt[2]*q^2) + 8*I*p*(-125 + 536*Sqrt[2]*q - 1320*q^2 + 352*Sqrt[2]*q^3))*Conjugate[q] + 16*(187 + 1216*p^4 + 128*I*p^3*(25*Sqrt[2] - 38*q) - 1072*Sqrt[2]*q + 4240*q^2 - 3200*Sqrt[2]*q^3 + 1216*q^4 + 16*p^2*(-265 + 600*Sqrt[2]*q - 456*q^2) - 16*I*p*(67*Sqrt[2] - 530*q + 600*Sqrt[2]*q^2 - 304*q^3))*Conjugate[q]^2 - 64*(41*Sqrt[2] + 192*Sqrt[2]*p^4 - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4 - 64*I*p^3*(-17 + 12*Sqrt[2]*q) - 32*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) + 8*I*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3))*Conjugate[q]^3 + 64*(17 + 64*p^4 + 64*I*p^3*(3*Sqrt[2] - 4*q) - 88*Sqrt[2]*q + 304*q^2 - 192*Sqrt[2]*q^3 + 64*q^4 + 16*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*I*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3))*Conjugate[q]^4 + 64*Conjugate[p]^3*(-192*I*Sqrt[2]*p^4 - 64*p^3*(-17 + 12*Sqrt[2]*q) + 32*I*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) + 8*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3) - I*(41*Sqrt[2] - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4) - 4*(-64*I*p^4 + 64*p^3*(3*Sqrt[2] - 4*q) - 16*I*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3) + I*(-17 + 88*Sqrt[2]*q - 304*q^2 + 192*Sqrt[2]*q^3 - 64*q^4))*Conjugate[q]) + 16*Conjugate[p]^2*(-187 - 1216*p^4 - 128*I*p^3*(25*Sqrt[2] - 38*q) + 1072*Sqrt[2]*q - 4240*q^2 + 3200*Sqrt[2]*q^3 - 1216*q^4 + p^2*(4240 - 9600*Sqrt[2]*q + 7296*q^2) + 16*I*p*(67*Sqrt[2] - 530*q + 600*Sqrt[2]*q^2 - 304*q^3) + 12*(41*Sqrt[2] + 192*Sqrt[2]*p^4 - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4 - 64*I*p^3*(-17 + 12*Sqrt[2]*q) - 32*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) + 8*I*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3))*Conjugate[q] + 24*(-17 - 64*p^4 - 64*I*p^3*(3*Sqrt[2] - 4*q) + 88*Sqrt[2]*q - 304*q^2 + 192*Sqrt[2]*q^3 - 64*q^4 + p^2*(304 - 576*Sqrt[2]*q + 384*q^2) + 8*I*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3))*Conjugate[q]^2) + 8*Conjugate[p]*(704*I*Sqrt[2]*p^4 + 704*p^3*(-5 + 4*Sqrt[2]*q) - 32*I*p^2*(67*Sqrt[2] - 330*q + 132*Sqrt[2]*q^2) - 8*p*(-125 + 536*Sqrt[2]*q - 1320*q^2 + 352*Sqrt[2]*q^3) + I*(81*Sqrt[2] - 1000*q + 2144*Sqrt[2]*q^2 - 3520*q^3 + 704*Sqrt[2]*q^4) + 4*(-1216*I*p^4 + 128*p^3*(25*Sqrt[2] - 38*q) - 16*I*p^2*(-265 + 600*Sqrt[2]*q - 456*q^2) - 16*p*(67*Sqrt[2] - 530*q + 600*Sqrt[2]*q^2 - 304*q^3) + I*(-187 + 1072*Sqrt[2]*q - 4240*q^2 + 3200*Sqrt[2]*q^3 - 1216*q^4))*Conjugate[q] + 24*(192*I*Sqrt[2]*p^4 + 64*p^3*(-17 + 12*Sqrt[2]*q) - 32*I*p^2*(25*Sqrt[2] - 102*q + 36*Sqrt[2]*q^2) - 8*p*(-55 + 200*Sqrt[2]*q - 408*q^2 + 96*Sqrt[2]*q^3) + I*(41*Sqrt[2] - 440*q + 800*Sqrt[2]*q^2 - 1088*q^3 + 192*Sqrt[2]*q^4))*Conjugate[q]^2 + 32*(-64*I*p^4 + 64*p^3*(3*Sqrt[2] - 4*q) - 16*I*p^2*(-19 + 36*Sqrt[2]*q - 24*q^2) - 8*p*(11*Sqrt[2] - 76*q + 72*Sqrt[2]*q^2 - 32*q^3) + I*(-17 + 88*Sqrt[2]*q - 304*q^2 + 192*Sqrt[2]*q^3 - 64*q^4))*Conjugate[q]^3))/E^(2*Abs[-(1/Sqrt[2]) + I*p + q]^2)/((Sqrt[2] - 4*I*p - 4*q)^2*(Sqrt[2] + 4*I*Conjugate[p] - 4*Conjugate[q])^2));


Go to polar coordinates by setting $$p=r\cos\phi$$ and $$q=r\sin\phi$$ and simplifying:

f[r_, ϕ_] = wig2[r Cos[ϕ], r Sin[ϕ]] // ComplexExpand // FullSimplify
(*    E^(-1 - 2 r^2 + 2 Sqrt[2] r Sin[ϕ]) *
(97 + 736 r^2 + 64 r^4 + 8 r (-34 r Cos[2 ϕ] - 5 Sqrt[2] (13 + 8 r^2) Sin[ϕ])) /
(81 π)    *)


Integrate numerically (without forgetting the Jacobian):

NIntegrate[Abs[f[r, ϕ]] r, {r, 0, ∞}, {ϕ, 0, 2 π}, Method -> "LocalAdaptive"]
(*    0.6804    *)


confirming @MariuszIwaniuk's result.

• This worked! Thanks a lot! Just wanted to know if there is any way to compute this integral analytically? Commented Aug 21, 2023 at 7:32
• @Anaya it's easy if you get rid of the Abs function, or use Abs[...]^2. But simply with Abs, as you have it, it requires quite a few case distinctions. I'll give it a try. Commented Aug 21, 2023 at 12:44
• The $Abs$ function is part of the formula that I am using to compute my answer, so it's not possible to remove it. Thanks a lot for your help Commented Aug 22, 2023 at 6:50