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# 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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