6

I am sorry if this question is a complete nonsense, but keep in mind that I am a senior in high school, so my math knowledge is really low.

My question is, can you graph complex zeroes on a three dimensional graph, where x and y axis would be real axis, and z axis would be imaginary one? Or is there another method of graphing those kinds of polynomials, which have one or more conjugated imaginary zeroes?

(I did read a bit about complex analysis, I am not sure if this falls under that branch of mathematics)

Murg
  • 250
  • 1
    Do you want go graph only zeros, or the polynomial itself? To graph zeros, you don't even need three dimensions - you can graph them as points on plane, with one coordinate been real part of zero and the other been imaginary. If you want to graph polynomial itself, you need four dimensions, as both variable and value of polynomial are two dimensional. – mihaild May 22 '19 at 18:10
  • How does that graph look? Are there four axis, or is it impossible to draw it, since we only have three dimensions? – Murg May 22 '19 at 18:14
  • It's impossible to draw it completely. There are ways to get around it - for example, draw real and imaginary parts separately, or colorize three dimensional graph. – mihaild May 22 '19 at 18:26
  • 1
    See also https://math.stackexchange.com/questions/607436/what-do-polynomials-look-like-in-the-complex-plane – lhf May 22 '19 at 18:37

1 Answers1

6

You can graph a complex function in the plane, using colours. This technique is called domain coloring.

Here is the graph of $f(z) = \dfrac{(z^2-i)(z-2-i)^2}{z^2+2+2i}$. Simple zeros have the rainbow once around it. Double zeros have the rainbow twice around it. The same for poles, but in reverse order.

I strongly recommend the book Visual Complex Functions by Wegert.

enter image description here

(image from Wikipedia)

lhf
  • 221,500