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Re: More strange behavior by ComplexExpand

• To: mathgroup at smc.vnet.net
• Subject: [mg60621] Re: More strange behavior by ComplexExpand
• From: Peter Pein <petsie at dordos.net>
• Date: Thu, 22 Sep 2005 02:08:14 -0400 (EDT)
• References: <dgr3c0$8g2$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Raul Martinez schrieb:
> 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.  
No, ComplexExpand treats your expression as complex. Have a look at

(a/Pi)^(1/4)*Exp[-(a*t^2)/2] /. a->-1

(0.5311259660135985 + 0.5311259660135984*I)*E^(0.5*t^2)

> This is puzzling to say the least. Moreover, it renders a^(1/4) as  
> (a^2)^(1/8), which seems bizarre.

((-1)^2)^(1/8) is easyer to handle than (-1)^(1/4).
ComplexExpand _treats_ a as real and gets rid of the negative case.

> 
> 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.
> 

Well, I guess you are thinking of

FullSimplify[ComplexExpand[(a/Pi)^(1/4)*Exp[-(a*t^2)/2]], a > 0]

a^(1/4)/(Sqrt[E^(a*t^2)]*Pi^(1/4))


> I welcome comments and suggestions.
> 
> Thanks in advance,
> 
> Raul Martinez
> 
> 
> 
> 
> 
> 


-- 
Peter Pein, Berlin
GnuPG Key ID: 0xA34C5A82
http://people.freenet.de/Peter_Berlin/