1

Inspired from:

"The Egg:" Bizarre behavior of the roots of a family of polynomials.

And much more information is at:

Wolfram Functions 3D plots over the complex planes for $z^a$

I came to notice that a surface plot of $$z=(x+yi)^n$$ had a saddle or a singularity made n lines of symmetry from the origin. Here are a few plots of the real part for simplicity:

Positive n: Notice the “saddle” shape and n lines of symmetry:

$n=3$:

enter image description here

$n=30$:

enter image description here

$n=2\pi$:

enter image description here

Negative n Notice the essential singularity at x=y=0 and n lines of symmetry:

$n=-3$:

enter image description here

$n=-10$:

enter image description here

$n=-3.7$:

enter image description here

$|n|<1$:: The closer to 0 the fraction is, the more circular the plot looks. The closer to 1 that n is, then the more it looks like a plane. A similar concept applies for $1<n<2$ and $-2<n<1$:

$n=\frac12$:

enter image description here

$n=-\frac12$:

enter image description here

Complex n: There seems to be no clear pattern other than a complex amount of “lines of symmetry”. We can also use De Moivre’s formula to split z into real and imaginary parts. Notice the oscillation:

$n=4i$:

enter image description here

$n=-2-3i$:

enter image description here

$n=-1-4i$:

enter image description here

$n=\frac1e-\frac1{\sqrt2}i:$

enter image description here

$n=6+i$:enter image description here

I already have some observations, but I would like to know what a graph of $$z=(x+yi)^n$$ would look like for a given n. For example, why does each bigger value of $|n|\in\Bbb R$ add more lines of symmetry and why does $n\in\Bbb C$ have such peculiar oscillatory behavior? I am guessing it has to do with the Riemann Surfaces of $z^n$. Also note that the argument of z will give insights as well. Please correct me and give me feedback!

The mystery of the symmetry has been solved by @Thomas Andrews, but behavior for complex and imaginary is still an open problem. If you have a better proof for real n than @Thomas Andrews, then please feel free to write it.

Тyma Gaidash
  • 12,081
  • 6
    If $\zeta_n=\cos(2\pi/n)+i\sin(2\pi/n)$ then $\zeta_n^n=1.$ sMultiplication by $\zeta_n$ rotates a complex number by $\frac{2\pi}{n}.$ And if $f(z)=z_n,$ then $f(\zeta_n z)=f(z),$ so you expect an $n$-fold rotational symmetry in the graph of $f(x+yi).$ – Thomas Andrews Sep 01 '21 at 01:52
  • @ThomasAndrews Is there any interpretation of the n-fold symmetry of $\mathrm{\frac{full\ turn}{power}=\frac{2\pi}{n}}$ for $n\in\Bbb C$? and not just real-n fold symmetry? It seems like hyperbolic trigonometric functions are involved for pure imaginary n. – Тyma Gaidash Sep 01 '21 at 02:14
  • 4
    It isn’t even possible to make $z^n$ continuous on all of $\mathbb C\setminus {0}$ if $n$ is not an integer. – Thomas Andrews Sep 01 '21 at 02:35
  • @ThomasAndrews Yes for example for $n\in \Bbb C, n=a+bi;a,b\in\Bbb R$ assuming a principal branch: $\mathrm{(x+yi)^{a+bi}=(x+yi)^a ((x+yi)^i)^b=(x+yi)^ae^{bi,ln(x+yi)}=(x+yi)^a [cos(b,ln(x+yi))+i,sin(b,ln(x+yi))]=\sqrt{x^2+y^2}e^{i,tan^{-1}\frac yx}+ [cos(b,ln(x+yi))+i,sin(b,ln(x+yi))]}$ Please also see this similar post. Now all that is left is to find ln(x+yi) seen Re and Im parts of ln(z).. Is this good? – Тyma Gaidash Sep 01 '21 at 12:50

0 Answers0