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Re: Plot of (2 x^2 - x^3)^(1/3)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112084] Re: Plot of (2 x^2 - x^3)^(1/3)
  • From: Peter Breitfeld <phbrf at t-online.de>
  • Date: Sun, 29 Aug 2010 02:51:27 -0400 (EDT)
  • References: <i5aqbk$eru$1@smc.vnet.net>

Bernard wrote:

> I see in Calcul Diff=E9rentiel et int=E9gral, N. Piskounov, Editions 
> MIR, Moscou 1970, on p. 210 the graph of (2 x^2 - x^3)^(1/3) with 
> negative values for x > 2, something like :
>
> Plot[Piecewise[{{(2 x^2 - x^3)^(1/3), x <= 2}, {-Re[(2 x^2 - 
> x^3)^(1/3)],  x > 2}}], {x, -1, 3}]
>
> When I plot this function with :
>
> Plot[(2 x^2 - x^3)^(1/3) // Re, {x, -1, 3}]
>
> I obtain positives values for x > 3. I don't understand why.
> Thank you very much for your help !

The reason is, that Mathematica always uses complex arithmetic.
Especially for multi-valued roots it takes the main value e^x=Exp[e Log[b]]

f[x_]:=(2x^2-x^3)^(1/3)

f[3]  --> (-1)^(1/3)*3^(2/3)
val=ComplexExpand[f[3]] --> (3/2)I*3^(1/6) + 3^(2/3)

So f[3] is a complex number with positive real part.

There are two ways to get your graph:

1. avoid non integer Powers, e.g. using ContourPlot
   ContourPlot[2x^2-x^2==y^3, {x,-1,3},{y,-3,3}]

2. define a version of Power, that uses the real branch like this:   

rprule=(b_?Negative)^Rational[m_,n_?OddQ]:>(-(-b)^(1/n))^m;
Attributes[realPower]={Listable,NumericFunction, OneIdentity};
realPower[b_?Negative,Rational[m_,n_?OddQ]]:=(-(-b)^(1/n))^m
realPower[x_,y_]:=x^y
realPower[x]:=x//.rprule

The last definition ensures, that realPower[(-8)^(1/3)] gives -2

Now Plot will work:
g[x_]:=realPower[2x^2-x^2,1/3]
Plot[g[x],{x,-1,3}]


-- 
_________________________________________________________________
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de


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