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Visualizing Functions of Complex Variables

Date: 05/09/2012 at 20:50:28
From: Maria
Subject: Functions of complex variables

Hi Dr. Math, 

Hope you are well.

I would like to ask you if you can explain to me how to visualize
functions of complex variables.

I am doing a course on complex analysis. I understand most theorems and
the analysis, but my teacher hasn't explained the basics so well. Are
functions of complex variables defined on the z plane, or are they
something like 4-dimensional?

Also, how can the integral over two homotopic paths be the same? I can't
picture that.

Wishing you all the best and thanking you in anticipation, 

Date: 05/11/2012 at 13:59:54
From: Doctor Tom
Subject: Re: Functions of complex variables

Hi Maria,

Yes, you're right: to "properly" visualize what's going on, you have to
think in four dimensions. Most of us aren't very good at doing that; so
here's how I think about it ...

If the function only yielded a real number output for each complex input,
visualization wouldn't be too hard. For every point in the complex plane,
place a point above (or below, if the value is negative) at a distance
equal to that real number function output. What you will obtain is a
surface above (and below) the plane. It's exactly the same as the way you
would visualize a real function of two variables: f(x, y). You plot the
output of f over (or under) the point (x, y) on the plane at a distance
equal to f(x, y).

For complex numbers, think of them as x + iy and do the same.

But the output is almost always a complex one -- not just a real number --
so one way to think of w = f(z) is as two different surfaces plotted above
the complex plane. One surface represents the real part of w; and the
other, the imaginary part of w. The result is just two 3-dimensional

Another, sometimes useful, way to visualize what's going on is to plot
various real-valued functions of f(z). A very useful one is |f(z)|. This
sort of tells you how "big" the output is. Less common might be to plot
arg(f(z)), the angle the point f(z) makes with the origin.

Yet another interesting way to visualize complex functions is to imagine a
picture or grid or something drawn on the plane, and to plot the distorted
version of that image on a plane after a function f acts on it. (You do
this all the time, but probably don't realize it: almost all standard map
projections, like the Mercator projection, are just analytic functions,
and the map of the earth on a flat map is exactly how the Mercator
projection works on the Riemann sphere with a map drawn on it.)

Finally: at least for me, it is NOT at all obvious why integrals over
homotopic paths are the same. That's a wonderful, amazing theorem from
complex analysis, and to me, it's amazing because it is totally
non-obvious. (Maybe for folks who work in complex variables all the time
it's more obvious, but I never got to that stage despite earning a PhD in
math from a pretty good school ...)

I hope this helps!

- Doctor Tom, The Math Forum 
Associated Topics:
College Analysis

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