Cube Root

Author

Eric W. Weisstein
January 2, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/CubeRoot.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CubeRoot.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Plot`

Real Axis

Plot[x^(1/3),{x,0,2},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,x^(1/3)}]

Complex  Plane

ComplexPlot3D[z^(1/3),{z,-5,5},ComplexPlotLabel->HoldForm[z^(1/3)]]//Timing

ComplexContourPlot[z^(1/3),{z,-5,5},ComplexPlotLabel->HoldForm[z^(1/3)]]//Timing

Roots and NRoots

Roots[x^3==-1,x]

N[%]

NRoots[x^3==-1,x]

Roots[x^3==1,x]

NRoots[x^3==1,x]

ComplexExpand[Power[-1,1/3]]

Iteration

RSolve[{x[n]==(2x[n-1]+a/x[n-1]^2)/3,x[0]==a},x[n],n]

With[{a=1.5,x0=0.5},{(a^(1/3)),FixedPoint[(1/3)(2#+a/#^2)&,x0]}]

With[{a=7.2123,x0=2},{(a^(1/3)),FixedPoint[(1/3)(2#+a/#^2)&,x0]}]

With[{a=7.,x0=-999},{(a^(1/3)),FixedPoint[(1/3)(2#+a/#^2)&,x0]}]