Re: visualization of f: C -> C

From: Larry Hammick (larryhammick_at_OMIT-MEtelus.net)
Date: 03/27/05 

Date: Sun, 27 Mar 2005 11:55:30 GMT

"Sukjah Roh"
> Hi
>
> I am sorry that I am not good at english.
> I am wondering how most mathematicians and some famous mathematicians
> visualize functions from C to C. (where C is the complex plane.)
>
> To make pictures in my mind, of functions of type R -> R, R->R^2,
> R^2->R, I imagine the graph of the functions. (the graph of a function
> R->R is in R^2, the graph of a function: R^2-> R is in R^3, etc)
>
> When it comes to functions of type C->C, the graph is in R^4 the 4
> dimensional space. so it is difficult to imagine the graph.
An analytic function is _conformal_ (meaning it preserves angles) wherever
its derivative is nonzero. Therefore it maps circles into circles, in the
sense that a line is a special kind of circle. This provides a way to
diagram complex functions by a sort of coordinate system, consisting of two
families of perpendicular curves. Lars Ahlfors has a pretty good account of
this topic in his book "Complex Analysis".
LH