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What is complex analysis?

This is one way to continue studying Calculus after Calc III. Calc I and II can be called "the theory of functions of one real variable." This course allows that variable to take complex values, by admitting the square root of -1 as a bona fide number. One might expect this theory to be basically identical, with minor adjustments here and there for "i", but in fact it differs in very significant ways, and often feels like a completely different branch of math. The reason for this change can be roughly explained: In order for a function f from the complex numbers (x+iy) to the complex numbers (u+iv), to be differentiable, of course four partial derivatives (du/dx, du/dy, dv/dx, dv/dy) must exist. But multiplication by i rotates the u-coordinate into the v-coordinate, and the v-coordinate into the -u-coordinate (and similarly for x and y). Because multiplication by i (or any other complex number) must commute (more or less) with the process of taking derivatives at a point, these four partials have to "match up when rotated". This statement amounts to the two equations
and , called the Cauchy-Riemann equations. These two equations together imply that complex-differentiable functions are very special. In particular, they preserve angles (which makes them useful in modelling fluid flow), and have harmonic real parts (which makes them great models for temperature fields).

One difficulty of reasoning about functions from the complex plane to the complex plane is that their graphs should be drawn on a four-dimensional space (two for the domain, and two for the range), but that's impossible.

Modern physics gives us strong empirical reasons to believe that imaginary and complex numbers do indeed exist, whatever that means. The tools developed in this course have very "real" applications, especially in quantum physics, despite our tendency to call them "imaginary"!

This page last modified Sun May 29 13:18:09 2005
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