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Re: More strange behavior by ComplexExpand

• To: mathgroup at smc.vnet.net
• Subject: [mg60615] Re: More strange behavior by ComplexExpand
• From: "dennis" <dwangsness at earthlink.net>
• Date: Thu, 22 Sep 2005 02:08:09 -0400 (EDT)
• References: <dgr3c0$8g2$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Raul Martinez wrote:
> To Mathgroup,
>
> I use Mathematica 5.2 with Mac OS X (Tiger).
>
> Add the following to a recent thread on the sometimes strange
> 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.
>
> I welcome comments and suggestions.
>
> Thanks in advance,
>
> Raul Martinez

Raul,

The problem is that a could be negative and still real, in which case
raising it to a fractional power results in a complex number.  I don't
think there is an error, you are just assuming a is positive.

Dennis