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Re: Explicit Conjugate: a feature or a bug?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg33899] Re: [mg33882] Explicit Conjugate: a feature or a bug?
*From*: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
*Date*: Mon, 22 Apr 2002 00:57:31 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
It seems to me that this may be due to the fact that Mathematica
considers Conjugate[(-1)^(1/3)] to be actually a simpler expression
than -(-1)^(2/3). Indeed, we have:
In[1]:=
LeafCount[Conjugate[(-1)^(1/3)]]
Out[1]=
6
In[2]:=
LeafCount[-(-1)^(2/3)]
Out[2]=
7
Mathematica tends to use LeafCount as the measure of complexity, which
sometimes does not correspond to the way humans perceive it. Thus:
In[3]:=
FullSimplify[Conjugate[(-1)^(1/3)]]
Out[3]=
Conjugate[(-1)^(1/3)]
but
In[4]:=
FullSimplify[Conjugate[(-1)^(1/3)], ComplexityFunction ->
(StringLength[ToString[#1]] & )]
Out[4]=
1/2 - (I*Sqrt[3])/2
and
In[5]:=
FullSimplify[Conjugate[(-1)^(1/3)], ComplexityFunction ->
(StringLength[ToString[#1]] & ),TransformationFunctions->{ComplexExpand}]
Out[5]=
-(-1)^(2/3)
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Sunday, April 21, 2002, at 07:12 PM, Vladimir Bondarenko wrote:
> Let's consider, say, Version 4.1 .
>
> Mathematically, the following is alright.
> On the other hand, the answer *explicitly* contains Conjugate.
>
> Integrate[((1 - z)/(-1 + I*z))^(1/3), {z, 0, 1}]
>
> (-I)*Conjugate[(-1)^(1/3)] + ((1 + I)^(4/3)*Conjugate[(-1)^(1/3)]*
> Hypergeometric2F1[1/3, 1/3, 4/3, 1/2 - I/2])/2^(1/3)
>
>
> But Conjugate[(-1)^(1/3)] looks VERY simply:
>
> Conjugate[(-1)^(1/3)]// ComplexExpand
>
> -(-1)^(2/3)
>
>
> Thus, Integrate[((1 - z)/(-1 + I*z))^(1/3), {z, 0, 1}] is just
>
> -(-1)^(1/6) - ((-1)^(2/3)*(1 + I)^(4/3)*
> Hypergeometric2F1[1/3, 1/3, 4/3, 1/2 - I/2])/2^(1/3)
>
> which, as for me, looks much nicer (but, of cause, has the same value).
>
> Is it a feature or a problem?
>
>
> Vladimir Bondarenko
>
>
> (* P.S. IMHO, this IS a bug because identifying -(-1)^(2/3) is
> trivial. *)
>
>
>
>
>
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