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I have a large Gaussian integral to evaluate, and Mathematica claims it does not converge:

Res = Integrate[Integrate[ 1/(32 \[Pi]^5)(E^(-1.881097845541816`Alpha1X^2 - 1.881097845541816` Alpha1Y^2-0.776219569108363` Alpha3X^2 - 0.776219569108363` Alpha3Y^2-1.6219812833697091` Alpha3X Alpha4X - 1.881097845541816` Alpha4X^2 +1.6219812833697087` Alpha3Y Alpha4Y - 1.8810978455418155` Alpha4Y^2 + 2.564577588805635` Alpha1Y AlphaUY - 0.7812669216517987` Alpha3Y AlphaUY + 2.293827928856761` Alpha4Y AlphaUY - 1.7429880609876338` AlphaUY^2 + 2.564577588805635` Alpha1X AlphaVX - 0.781266921651799` Alpha3X AlphaVX - 2.293827928856761` Alpha4X AlphaVX -1.7429880609876338` AlphaVX^2 - 2.564577588805635` Alpha1X xm +1.4792886705085173` Alpha3X xm + 2.9495448495343135` Alpha4X xm +3.4731279460398268` AlphaVX xm - 2.199924453259773` xm^2 -2.564577588805635` Alpha1Y ym + 1.479288670508518` Alpha3Y ym -2.9495448495343126` Alpha4Y ym + 3.4731279460398268` AlphaUY ym -2.1999244532597744` ym^2)), {Alpha3X, -\[Infinity], \[Infinity]},Assumptions -> {Im[xm] == 0,Im[ym] == 0}], {Alpha3Y, -\[Infinity], \[Infinity]}]

However, when I remove the sections of the exponent that are independent of the integration variable, it is able to converge to a value:

Res = 1/(32 \[Pi]^5)*Exp[-1.881097845541816` Alpha1X^2 - 1.881097845541816` Alpha1Y^2 - 1.881097845541816` Alpha4X^2 - 1.8810978455418155` Alpha4Y^2 + 2.564577588805635` Alpha1Y AlphaUY + 2.293827928856761` Alpha4Y AlphaUY - 1.7429880609876338` AlphaUY^2 + 2.564577588805635` Alpha1X AlphaVX - 2.293827928856761` Alpha4X AlphaVX - 1.7429880609876338` AlphaVX^2 - 2.564577588805635` Alpha1X xm + 2.9495448495343135` Alpha4X xm + 3.4731279460398268` AlphaVX xm - 2.199924453259773` xm^2 - 2.564577588805635` Alpha1Y ym - 2.9495448495343126` Alpha4Y ym + 3.4731279460398268` AlphaUY ym - 2.1999244532597744` ym^2]*Integrate[E^(-0.776219569108363` Alpha3X^2 - 0.776219569108363` Alpha3Y^2 - 1.6219812833697091` Alpha3X Alpha4X + 1.6219812833697087` Alpha3Y Alpha4Y - 0.7812669216517987` Alpha3Y AlphaUY - 0.781266921651799` Alpha3X AlphaVX + 1.4792886705085173` Alpha3X xm + 1.479288670508518` Alpha3Y ym), {Alpha3X, -\[Infinity],\[Infinity]}, {Alpha3Y, -\[Infinity], \[Infinity]}]


= 0.000413301 E^(-1.8811 Alpha1X^2 - 1.8811 Alpha1Y^2 - 1.03378 Alpha4X^2 - 1.03378 Alpha4Y^2 + 2.56458 Alpha1Y AlphaUY + 1.47756 Alpha4Y AlphaUY - 1.5464 AlphaUY^2 + 2.56458 Alpha1X AlphaVX - 1.47756 Alpha4X AlphaVX - 1.5464 AlphaVX^2 - 2.56458 Alpha1X xm + 1.40399 Alpha4X xm + 2.72867 AlphaVX xm - 1.49513 xm^2 - 2.56458 Alpha1Y ym - 1.40399 Alpha4Y ym + 2.72867 AlphaUY ym - 1.49513 ym^2)

A similar issue with a different integral happened on my desktop Mathematica version (13.3.1, Windows), and it was fixed by using the online cloud version (13.3.1, Linux), but now neither are able to converge. Sometimes the issue was fixed when I ran the line multiple times after each other, however that does not work now. When I tried this on an older version (12.2 Mac), it was able to do these.

Are there any additional assumptions I need to add to clarify this for the integral to evaluate? How come repeating the evaluation worked previously but doesn't now? Thanks in advance.

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    $\begingroup$ Your first input has syntax error. Missing closing ). When I corrected it, it gives this answer on V 13.3.1 with no error !Mathematica graphics btw, you should be using non-exact numbers with Integrate. Could cause some problems sometimes. $\endgroup$
    – Nasser
    Commented Oct 7, 2023 at 5:35
  • $\begingroup$ Ah sorry, that was a pasting error - correcting it still doesn't work on my versions output photos. $\endgroup$ Commented Oct 7, 2023 at 5:46
  • $\begingroup$ Well, it works for me on windows 10, V 13.3.1. It could be version issue. !Mathematica graphics but again, try to rationalize the integrand so numbers are exact first, may be that will help on your version. $\endgroup$
    – Nasser
    Commented Oct 7, 2023 at 5:57

3 Answers 3

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It works fine if you use assumptions properly: to declare that a variable has no imaginary part, assert that it is in the Reals. However, in the given example this assumptions makes no difference.

Assuming[Element[xm | ym, Reals],
  1/(32 π^5) Exp[
    -1.881097845541816` Alpha1X^2 - 1.881097845541816` Alpha1Y^2 -
    0.776219569108363` Alpha3X^2 - 0.776219569108363` Alpha3Y^2 -
    1.6219812833697091` Alpha3X Alpha4X - 1.881097845541816` Alpha4X^2 +
    1.6219812833697087` Alpha3Y Alpha4Y - 1.8810978455418155` Alpha4Y^2 +
    2.564577588805635` Alpha1Y AlphaUY - 0.7812669216517987` Alpha3Y AlphaUY +
    2.293827928856761` Alpha4Y AlphaUY - 1.7429880609876338` AlphaUY^2 +
    2.564577588805635` Alpha1X AlphaVX - 0.781266921651799` Alpha3X AlphaVX -
    2.293827928856761` Alpha4X AlphaVX - 1.7429880609876338` AlphaVX^2 -
    2.564577588805635` Alpha1X xm + 1.4792886705085173` Alpha3X xm +
    2.9495448495343135` Alpha4X xm + 3.4731279460398268` AlphaVX xm -
    2.199924453259773` xm^2 - 2.564577588805635` Alpha1Y ym +
    1.479288670508518` Alpha3Y ym - 2.9495448495343126` Alpha4Y ym +
    3.4731279460398268` AlphaUY ym - 2.1999244532597744` ym^2],
  {Alpha3X, -∞, ∞}, {Alpha3Y, -∞, ∞}]]

(*    0.000413301 E^(-1.8811 Alpha1X^2 - 1.8811 Alpha1Y^2 -
      1.03378 Alpha4X^2 - 1.03378 Alpha4Y^2 +
      2.56458 Alpha1Y AlphaUY + 1.47756 Alpha4Y AlphaUY -
      1.5464 AlphaUY^2 + 2.56458 Alpha1X AlphaVX -
      1.47756 Alpha4X AlphaVX - 1.5464 AlphaVX^2 -
      2.56458 Alpha1X xm + 1.40399 Alpha4X xm +
      2.72867 AlphaVX xm - 1.49513 xm^2 -
      2.56458 Alpha1Y ym - 1.40399 Alpha4Y ym +
      2.72867 AlphaUY ym - 1.49513 ym^2)                    *)
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Mathematica is becoming more exact and more complicated to use with each version.

Gaussian integrals with symbols do not converge, except the real part of the square in the exponent is positive, such that $$ \Re a >0 \longrightarrow \lim_{x\to \pm \infty} e^{- a x ^2 + \text{linear terms}} =0$$

In your case, replace your non evaluating

    Res = Res/. Integrate :>Inactive[Integrate]

gather the symbols

 smb = Cases[Res, Except[ E | \[Pi], _Symbol], \[Infinity]] // Union;

and integrate with all symbols restricted to reals

Assuming[smb \[Element] Reals, Activate[Res]]

$$\begin{align}&0.000413301\\& * \ \exp(-1.8811 Alpha1X^2 - 1.8811 Alpha1Y^2 \\&- 1.03378 \ Alpha4X^2 - 1.03378 \ Alpha4Y^2 + 2.56458 \ Alpha1Y\ AlphaUY \\&+ 1.47756 \ Alpha4Y \ AlphaUY - 1.5464 \ AlphaUY^2 + 2.56458 \ Alpha1X \ AlphaVX - 1.47756 \ Alpha4X \ AlphaVX \\&- 1.5464 \ AlphaVX^2 - 2.56458 \ Alpha1X \ xm + 1.40399 \ Alpha4X\ xm \\&+ 2.72867 \ AlphaVX \ xm - 1.49513 \ xm^2 - 2.56458 \ Alpha1Y \ ym - 1.40399 \ Alpha4Y \ ym + 2.72867 \ AlphaUY \ ym - 1.49513 \ ym^2) \end{align}$$

A shortcut for such problems is

 Assuming[ _ \[Element] Reals, Res ]
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Mathematica v12.2. evaluates without any assumptions:

 Integrate[
 1/(32 \[Pi]^5) (E^(-1.881097845541816` Alpha1X^2 - 
      1.881097845541816` Alpha1Y^2 - 0.776219569108363` Alpha3X^2 - 
      0.776219569108363` Alpha3Y^2 - 
      1.6219812833697091` Alpha3X Alpha4X - 
      1.881097845541816` Alpha4X^2 + 
      1.6219812833697087` Alpha3Y Alpha4Y - 
      1.8810978455418155` Alpha4Y^2 + 
      2.564577588805635` Alpha1Y AlphaUY - 
      0.7812669216517987` Alpha3Y AlphaUY + 
      2.293827928856761` Alpha4Y AlphaUY - 
      1.7429880609876338` AlphaUY^2 + 
      2.564577588805635` Alpha1X AlphaVX - 
      0.781266921651799` Alpha3X AlphaVX - 
      2.293827928856761` Alpha4X AlphaVX - 
      1.7429880609876338` AlphaVX^2 - 2.564577588805635` Alpha1X xm + 
      1.4792886705085173` Alpha3X xm + 
      2.9495448495343135` Alpha4X xm + 
      3.4731279460398268` AlphaVX xm - 2.199924453259773` xm^2 - 
      2.564577588805635` Alpha1Y ym + 1.479288670508518` Alpha3Y ym - 
      2.9495448495343126` Alpha4Y ym + 
      3.4731279460398268` AlphaUY ym - 
      2.1999244532597744` ym^2)), {Alpha3X, -\[Infinity], \
\[Infinity]}, {Alpha3Y, -\[Infinity], \[Infinity]}]

    (* 0.000413301 E^(-1.8811 Alpha1X^2 - 1.8811 Alpha1Y^2 - 
  1.03378 Alpha4X^2 - 1.03378 Alpha4Y^2 + 2.56458 Alpha1Y AlphaUY + 
  1.47756 Alpha4Y AlphaUY - 1.5464 AlphaUY^2 + 
  2.56458 Alpha1X AlphaVX - 1.47756 Alpha4X AlphaVX - 
  1.5464 AlphaVX^2 - 2.56458 Alpha1X xm + 1.40399 Alpha4X xm + 
  2.72867 AlphaVX xm - 1.49513 xm^2 - 2.56458 Alpha1Y ym - 
  1.40399 Alpha4Y ym + 2.72867 AlphaUY ym - 1.49513 ym^2)*)
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