More strange behavior by ComplexExpand

• To: mathgroup at smc.vnet.net
• Subject: [mg60603] More strange behavior by ComplexExpand
• From: Raul Martinez <raulm231 at comcast.net>
• Date: Wed, 21 Sep 2005 03:20:48 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

To Mathgroup,

I use Mathematica 5.2 with Mac OS X (Tiger).

behavior of ComplexExpand.

I used ComplexExpand with an argument in which all the variables in
the argument of the function are real. Since ComplexExpand is
supposed to assume that all variables are real by default, one would
expect ComplexExpand to return the expression without change, but it
doesn't. Instead, here is what it does:

In[1]:=

ComplexExpand[ (a / Pi)^(1/4) Exp[ (-(a t^2)/2 ] ]

Out[2]:=

(Exp[-a t^2] Sqrt[Exp[a t^2]] (a^2)^(1/8) Cos[Arg[a] / 4]) / Pi^
(1/4) + i (Exp[-a t^2] Sqrt[Exp[a t^2]] (a^2)^(1/8) Sin[Arg[a] /
4]) / Pi^(1/4).

I have inserted parentheses in a few places to improve the legibility
of the expressions.

ComplexExpand treats the variable "a" as complex, but "t" as real.
This is puzzling to say the least. Moreover, it renders a^(1/4) as
(a^2)^(1/8), which seems bizarre.

My interest is not in obtaining the correct result, which is easy to
do. Rather, I bring this up as yet another example of the
unreliability of ComplexExpand. In case anyone is wondering why I
would use ComplexExpand on an expression I know to be real, the
reason is that the expression in question is a factor in a larger
expression that contains complex variables. Applied to the larger
expression, ComplexExpand returned an obviously incorrect expansion
that I traced to the treatment of the example shown above.