# Convergence of integration - test

Question: Is there a way to evaluate convergence for complicated integrals in Mathematica?

I have a 12 dimensional integral - given by the following code

q3v = Table[q3[i], {i, 3}];
p3v = Table[p3[i], {i, 3}];
qv = Table[q[i], {i, 3}];
pv = Table[p[i], {i, 3}];
intvariables = Flatten[Join[{q3v}, {p3v}, {qv}, {pv}]]
intvariables2 = ({#1, -1, 1} & ) /@ intvariables

a := 1/(Sqrt[m^2 + pv . pv]*Sqrt[m^2 + p3v . p3v]*
Sqrt[μ^2 + q3v . q3v]*
Sqrt[μ^2 + (pv - p3v - q3v) . (pv - p3v - q3v)]);
b := 1/(Sqrt[m^2 + p3v . p3v]*Sqrt[m^2 + qv . qv]*
Sqrt[μ^2 + q3v . q3v]*
Sqrt[μ^2 + (-p3v + qv - q3v) . (-p3v + qv - q3v)]);
c := 1/(Sqrt[μ^2 + q3v . q3v] +
Sqrt[μ^2 + (-p3v - q3v + qv) . (-p3v - q3v + qv)]);
e := 1/(pv . pv + 1^2)^2;
f := 1/(qv . qv + 1^2)^2;


and then

jj = FullSimplify[a*b*c];
TraditionalForm[jj]


[Comment: if I try FullSimplify[a b c e f], Mathematica 10 and 11 just hang..]

If I now try

jjm = jj //. {μ -> 1., m -> 1.}


followed by

NIntegrate[jjm*e*f, Evaluate[Sequence @@ intvariables2], MaxPoints -> 1000]


the result is -22.54358093170731

However, if the integration is done as

intvariables3 = ({#1, -100, 100} & ) /@ intvariables
NIntegrate[jjm*e*f, Evaluate[Sequence @@ intvariables3], MaxPoints -> 1000]


gives a completely different result

NIntegrate::maxp: The integral failed to converge after 13227 integrand evaluations. NIntegrate obtained -3.78792*10^23 and 1.2256724176744906*^24 for the integral and error estimates. >>

1. Is it normal for Mathematica to hang - for the command jj = FullSimplify[a*b*c*e*f]; TraditionalForm[jj]`

2. How does one test for convergence of integration, given that Mathematica won't integrate jjm symbolically, between -Infinity and Infinity.

• Please put a notice in the question when cross-posted: community.wolfram.com/groups/-/m/t/1177659 – Michael E2 Sep 7 '17 at 11:42
• I deleted my answer, because there was something wrong with the usage of Max. I try to imporve it. – Akku14 Sep 9 '17 at 10:50