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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