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, Maria
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 surfaces. 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 http://mathforum.org/dr.math/
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