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Re: Re: branch of (-1)^(1/3)


slawek wrote:

>U¿ytkownik "Carl Woll" <carlw at wolfram.com> napisa³ w wiadomo¶ci 
>news:gi2uug$a6m$1 at smc.vnet.net...
>  
>
>>I assume you mean a simple way to choose the branch of a^(1/3), where a
>>is real. If so, you can use:
>>
>>Root[#^3-a&, 1]
>>    
>>
>
>
>It doesn't work, because this appoach may be used in this way
>
>In[24]:= (-1)^(1/3) /. a_^(1/3) -> Root[#^3 - a &, 2]
>Out[24]= 1/2 (1 - I Sqrt[3])
>
>nevertheless is completly unusable in this example
>
>In[25]:= (-Sin[x])^(1/3) /. a_^(1/3) -> Root[#^3 - a &, 2]
>Out[25]= Root[Sin[x] + #1^3 &, 2]
>  
>
Why is it unusable? For example,

Plot[Root[Sin[x] + #^3 &, 1], {x, -Pi, Pi}]

generates a nice plot. Just because the Root object only simplifies when 
x takes on special values doesn't mean there is anything wrong with it.

Carl Woll
Wolfram Research

>You can see, that Root works only for "numbers" - whereas a simple Sin[x] is 
>enouch to stop evaluating Root[ ].
>
>Obviously, I still can make the calculation by the pencil... and maybe my 
>old log ruler. Is Mathematica suitable for?
>
>And there is no help if
>
>something /. a_^(1/3) -> Abs[a]^(1/3)
>
>because the ^(1/3) still will fail recognize that the Abs[a] is real and 
>that most obvious is a real result. (Because there are many things that are 
>really real - for example taxes - if in any country the tax will be computed 
>as (income/factor)^(-1/3) it would mean that the tax is imaginary!!! :) )
>
>Regarding the numbering of roots: the system is arbitrary, it may be any 
>permutation and the mathematics will remain the same.
>The real - non-real is real: we are in R or in C, quite different sets.
>The "branch number" is artifical similary as artifical are plate numbers on 
>cars.
>
>slawek
> 
>
>  


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