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Re: FullSimplify doesn't simplify

• To: mathgroup at smc.vnet.net
• Subject: [mg32768] Re: [mg32762] FullSimplify doesn't simplify
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Sat, 9 Feb 2002 05:11:38 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

Actually there are two different questions here. Let's start with the 
second.


In[9]:=
ComplexExpand[(-I)^I]

Out[9]=
E^(Pi/2)

In[10]:=
ComplexExpand[(-1)^I]

Out[10]=
E^(-Pi)

In[11]:=
ComplexExpand[I^I]

Out[11]=
E^(-Pi/2)


So (-1 * I)^I != (-1)^I * (I)^I

You can also verify this using FullSimplify:

In[12]:=
FullSimplify[(-1*I)^I==(-1)^I*(I)^I]

Out[12]=
False

However this is not the reason why Mathematica does not simplify in your 
case! To see that consider:

In[13]:=
FullSimplify[(x*y)^n,Element[n,Reals]]

Out[13]=
(x*y)^n

However the identity is indeed valid for real exponents.

The point is that Simplify (and FullSimplify) will not in general use 
transformations involving symbolic powers (except for a few of the 
simplest ones that are universally true and are usually applied even 
without without Simplify, e.g.
In[14]:=
(2*a)^y

Out[14]=
2^y*a^y
). The reason is that the general algorithms on which these 
transformations depend are essentially algebraic, which means they work 
on polynomials and rational functions and some special functions like 
trigonometric functions etc.


On Friday, February 8, 2002, at 05:49  PM, Ken Morgan wrote:

> Does anyone know why Mathematica doesn't simplify the following to True?
>
>
> In[1] := FullSimplify[(x y)^n == (x^n)(y^n)]
>
> Out[1] := (x y)^n == (x^n)(y^n)
>
> Are there cases where this is not true?
>
>
> Thanks,
> Ken Morgan
> kmorga51 at calvin.edu
>
>
>
>
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/