Almost all of modern physics relies on the use of complex
Quantum Mechanics: probability wavefunctions are complex quantities
Relativity: The tensor formulation of special relativity requires
the time variable to have a factor of i in order to
form the correct invariant metric for space-time.
Solid state physics: quantization of normal modes of vibration
lead naturally to the use of complex variables
And in fact: any wave equation requires complex variables for a full
Then there's math itself: many integrals can only be solved by
use of contour integration in the complex plane.
--- and the general solution of polynomial equations of course requires
complex numbers. In fact, the fundamental theorem of algebra states
that polynomial equations (quadratic, cubic, quartic, etc) are
completely solved under the set of complex numbers. (Aside: this
theorem is beautifully presented and explained in one of the
of Volume I of the Feynman Lectures on Physics. Along the way,
Feynman explains how to calculate any logarithm from a few known
values, and derives *numerically* the Euler identity relating
complex exponentials and trig functions; a remarkable 20-30 pages
Solution of electrostatics problems are often very easy by use of
conformal mapping, which maps a problem space by use of a function
of a complex variable to a simpler space that is easily solved, the
using the inverse function to map the solution back to the original
space. Very elegant!
I've barely scratched the surface. There is a book, History of
[sqrt(-1)] (using the actual symbols in the title, of course),
which mentions some of the above examples and many
many more. Written by an electrical engineer, so ought to be
accessible to most EE's.
Doug Brooks wrote:
> Complex algebra, of the form
> Z = R + jX
> is of course a very important part of electronics, which is where my
> background is.
> I am curious in what other disciplines do we find complex math? There must
> be many of them, but I have never come across any of them. I'd like to
> build up a sort of list of several examples.
> Can anyone help out?
> Doug Brooks
> Doug Brooks, President firstname.lastname@example.org
> UltraCAD Design, Inc. http://www.ultracad.com
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