# Bivariate Integrations/Assumptions error/

• To: mathgroup@smc.vnet.net
• Subject: [mg10603] Bivariate Integrations/Assumptions error/
• From: Mary Lesperance <mlespera@uvic.ca>
• Date: Mon, 26 Jan 1998 04:42:34 -0500
• Organization: University of Victoria

```Mathematica has trouble with bivariate integrations.  I tried the
following:  (The answer is 1 for any real Phi and Psi - this is just
the product of 2 normal densities)

f[x_,y_,\[Phi]_,\[Psi]_,n_]:=.5*Exp[-.5*((x-\[Phi])^2 + (y-(\[Phi]^2
+   \[Psi]))^2)]/Pi

Integrate[f[x,y,\[Phi],\[Psi],1],{x,-Infinity,Infinity},{y,-Infinity,Infinity},Assumptions
-> {Im[\[Phi]]==0 && Im[\[Psi]]==0}]

The answer that Mathematica returns starts with:

(If[\(Re[\(-1.`\)\ \[Phi]\^2 - 1.`\ \[Psi]] > 0 &&
Re[1.`\ \[Phi]\^2 + 1.`\ \[Psi]] > 0 && Re[\[Phi]] < 0, \)\)\)

i.e. a condition IF Re(Phi^2 + Psi)<0  AND  Re(Phi^2 + Psi)>0, then
resulting
answer;   Clearly, the assumptions state that this cannot be true.

if the condition is false, Mathematica returns the unevaluated integral.
---------------
Sometimes Mathematica returns the value def. integral ZERO - it depends
on
Mathematica's "state of mind"!    And sometimes it just shuts down
altogether:  The Kernel Local has quit(exited) during the course of an
evaluation.  Why?
----------------

The following seems to perform better - I've told it not to generate
conditions:

Integrate[f[x,y,\[Phi],\[Psi],1],{x,-Infinity,Infinity},{y,-Infinity,Infinity},Assumptions
-> {Im[\[Phi]]==0 && Im[\[Psi]]==0},GenerateConditions->False]

The answer is 1, except Mathematica cannot simplify to that level. After
Chop[%] and FullSimplify[%] it returns:
Exp^0 . Phi^2  .  (Phi^2 + Psi)^2 / (Phi^2 + Psi)^2

----------------
I am running this on an HP workstation, and am truly disappointed.
After spending big \$\$ on Mathematica, I expected better.
performs this computation in very little time.  I would appreciate
hints as to "teaching" Mathematica to perform these types of
integrations more smoothly.  I would like to progress to more difficult
expressions.  Thanks.

Mary Lesperance, University of Victoria

```

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