Vtyjkyuk

The graph of a complex function $f: mathbbC rightarrow mathbbC$ is a 3-dimensional object in a 4-dimensional space and thus hard to visualize, even when it's a smooth 3-dimensional surface.

A natural way to visualize it is by two graphs of two functions $r: mathbbR^2rightarrow mathbbR, i: mathbbR^2rightarrow mathbbR$ with $r(z) = textRe(f(z)), i(z) = textIm(f(z))$. These functions define - under appropriate circumstances - two 2-dimensional objects $colorredS^f_r$ and $colorgreenS^f_i$ in a 3-dimensional space and can be visualized in $mathbbR^3$, at least when they are smooth 2-dimensional surfaces.

In general these two surfaces may or may not

• intersect with each other at some points in $mathbbR^3$ giving a 1-dimensional object $colorblueC^f= colorredS^f_r cap colorgreenS^f_i$ which lives in $mathbbR^3$


• intersect with the plane $mathbbR^2$ giving two 1-dimensional objects $colorredC^f_r = colorredS^f_r cap mathbbR^2$ and $colorgreenC^f_i = colorgreenS^f_i cap mathbbR^2$ which live in $mathbbR^2$

These 1-dimensional objects can be straight lines (or sets of straight lines), circles (or sets of circles), arbitrary open or closed curves (or sets of those).

One thing is obvious: There are $z_0 in mathbbC$ with $f(z) = 0$ (i.e. $f$ has roots) iff $colorredC^f_r cap colorgreenC^f_i neq emptyset$. Knowing that each complex polynomial has roots (the fundamental theorem of algebra), we know that $colorredC^P_r cap colorgreenC^P_i neq emptyset$ for all polynomials $P$. (We know even more: $colorredC^P_r cap colorgreenC^P_i$ is a point set of size less or equal the degree of $P$.)

Example 1:

$P(z) = z^2 -1, r(x,y) = x^2 - y^2 - 1 , i(x,y) = 2xy$

$colorredC^P_r = x^2 - y^2 = 1 $, $colorgreenC^P_i = x = 0 vee y = 0 $

$P(z) = 0 leftrightarrow z = (1,0) vee z = (-1,0)$

Example 2:

$P(z) = z^2 + 1, r(x,y) = x^2 - y^2 + 1 , i(x,y) = 2xy$

$colorredC^P_r = (x,y) $, $colorgreenC^P_i = x = 0 vee y = 0 $

$P(z) = 0 leftrightarrow z = (0,1) vee z = (0,-1)$

What I'd like to know:

1. What can - beyond the fundamental theorem - be said about the shapes and the positions of $colorredC^P_r$ and
   $colorgreenC^P_i$ for polynomials $P$ in general terms (depending
   on the degree of $P$)? May (or even does) the fundamental theorem
   result from the characterizations of $colorredC^P_r$ and
   $colorgreenC^P_i$?



2. 
   

   How may holomorphic functions (= analytic complex functions) be characterized in terms of $colorblueC^f, colorredC^f_r, colorgreenC^f_i$?

   
   
   
   • Like this: "For a holomorphic function $f$ the set $colorblueC^f$ must be non-empty and such-and-such"?
   
   
   • Like that: "When a curve $C$ is such-and-such there is a holomorphic function $f$ with $C = colorblueC^f$"?
   
   
   • Might a
     holomorphic function $f$ possibly be fixed by its $colorblueC^f$
     alone or by the pair $(colorredC^f_r, colorgreenC^f_i)$?
   
   

For the learned mathematician the answers to these question may seem obvious ("How can you ask?"), for me they are not, sorry.

Riemann's Surfaces may be relevant in visualizing multi-dimensional (complex) functions. There is an excellent video series by welch labs in their playlist of Imaginary Numbers are Real.

Even though they start with the history and visualization of Complex functions, they move on to show 4-D Complex functions on 3-D Plane (not exactly the function, more like it's shadow) quite beautifully with computer animations. Take a look at it, it might help.