Why does not Maple plot $y$=$x^{1/3}$ properly?

Question

Many of my students have asked me this simple question:

"Why does not Maple plot $y$=$x^{1/3}$ properly?"

I have examined it by myself. Also, I ploted the absolute value of $y$ and think that, it is a defect in this maths tool. Is there any suggestion? Thanks.

Answer 1

What to you mean with a "proper" plot? As I can guess the problem is that Maple does not plot it for $x<0$. Usually $x^\alpha$ for $\alpha\notin \mathbb{N}$ is defined only for non-negative $x$.

I mean, that if we allow to calculate $(-1)^{1/3}$ then $$ -1 = (-1)^{1/3} = (-1)^{2/6} = (1)^{1/6}=1. $$

On the other hand, sometimes $x^{1/n}$ for an odd $n$ can be defined as a real root of $y^n = x$ - so this is a matter of convention.

Answer 2

If $z < 0$, Maple is using the principal branch of the log along with the identity $$z^{1/3} = \exp((1/3)*\log(z)).$$
If you use $z = -1$, you get $$z^{1/3} = \exp((1/3)*\log(z)) = \exp(1/3*\pi) = \exp(i\pi/3) = {1 + \sqrt{3}\over 2}.$$ In a word, Maple is being "scrupulous to a fault." It's a small price to pay for the program being so complex-number savvy.

Answer 3

I suspect Maple, like many other computer algebra systems, decides that $a^b$ for negative real $a$ and real $b$ to be something like $|a|^b \; e^{i \pi b}$.

This makes a lot of sense as a convention (it is continuous in $b$), even if it might be unexpected for $k$ as the reciprocal of a positive integer.

Answer 4

According to Maple, $x^{1/3}$ is a non-real complex number when $x<0$.
evalf((-1)^(1/3)) yields .5000000001+.8660254037*I

If you want to force the real cube root, use surd ... type ?surd for information.