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I was trying to plot a very simple polynomial function:

Plot[x^(2./3.),{x,-0.5,-0.2}]

Mathematica tells me that in this region, the result is always imaginary, but from fundamental math the result should be real. So why the difference?

To make the question clear: I know how to get the correct plot, but I want to understand why mathematica gives us imaginary result. How did mathematica do the calculation inside the kernal?

Thanks!

J. M.'s missing motivation
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Y.Du
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2 Answers2

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Plot[CubeRoot[x]^2, {x, -0.5, -0.2}]

or

Plot[Surd[x, 3]^2, {x, -0.5, -0.2}]
jiaoeyushushu
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  • Hi jiaoeyushushu, thanks for your reply. I actually know how to get the correct plot. But do you know why mathamatica gives us imaginary results? Thanks. – Y.Du Aug 30 '17 at 15:14
  • From the Documentation Center:

    Power[x,y] has a branch cut discontinuity for y running from -[Infinity] to 0 in the complex x plane for noninteger y. Because of this branch cut, Power[x,1/n] returns a complex root by default instead of the real one for negative real x and odd positive n. To obtain a real-valued n[Null]^th root, Surd[x,n] can be used. The special case CubeRoot[x] corresponds to Surd[x,3].

     – John Doty Aug 30 '17 at 15:25
  • @JohnDoty Thank you John and also corey979, your answers solve my question. – Y.Du Aug 30 '17 at 15:38
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Guide Mathematica to the desired result, 

Table[(x^(2.))^(1/3.), {x, -0.5, -0.2, 0.1}]

{0.629961, 0.542884, 0.44814, 0.341995}

If you check the internal operation procedure you will see the difference,

FullForm[x^(2./3.)]

enter image description here

compare with this

FullForm[(x^(2.))^(1/3.)]

enter image description here

You can also try with exact form, i.e., 2/3.

zhk
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