Can I get ComplexExpand to really work?

• To: mathgroup at smc.vnet.net
• Subject: [mg14536] Can I get ComplexExpand to really work?
• From: Topher Cawlfield <cawlfiel at uiuc.edu>
• Date: Thu, 29 Oct 1998 04:33:31 -0500
• Organization: University of Illinois at Champaign-Urbana
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

I'm having lots of problems getting Mathematica to make simplifying
assumptions.  It always seems to want to produce horribly complex
results because it assumes every variable is complex.  I wish I had
better control of that.  In fact, it would also be nice if I could
assure Mathematica that certain variables were positive as well.

It sounds like the function ComplexExpand should do the trick, at least
by assuming that variables are real unless otherwise specified.  But it
doesn't really seem to work for me.  Here's an example:

ComplexExpand[Integrate[E^(I a x^2), {x, -Infinity, Infinity}]]

produces:

If[Im[a] == 0, Sqrt[Pi/2] (1 + I Sign[a]) / (a^2)^(1/4), Integrate[E^(I
a x^2), {x, -Infinity, Infinity}]]

But if it really was assuming that 'a' was real, then it should know
that Im[a] == 0!  If I could also tell it that 'a' was positive, the

(1 + I) Sqrt[Pi/a]  or better still, Sqrt[2 Pi I / a]

This is much simpler, and is the answer I want.

Of course, my real application of this problem is much more complicated,
but ultimately comes down to doing that integral (several times over).
The right answer should be just about that simple, but instead
Mathematica gives me about 5 pages of output.

Is there any hope of getting reasonable symbolic results here?

- Topher Cawlfield

```

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