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Complex variables are often accepted in other parts
of analysis when this causes no essential change in the theory.
Here we consider those aspects of analytic behaviour unique to complex
functions. These functions and those used to describe phenomena in part of
mathematical physics both display a considerable
degree of regularity not found in general functions of a real variable.
30: Complex variables studies the effect of
assuming differentiability of functions defined on complex
numbers. Fascinatingly, the effect is markedly different than for real
functions; these functions are much more rigidly constrained, and in
particular it is possible to make very definite comments about their
global behaviour, convergence, and so on. This area includes Riemann
surfaces, which look locally like the complex plane but aren't the same
space. Complex-variable techniques have great use in applied areas
(including electromagnetics, for example).
31: Potential theory studies harmonic functions
(and their allies). Mathematically, these are solutions to the Laplace
equation Del(u)=0; physically, they are the functions giving the
potential energy throughout space resulting from some masses or electric
32: Several complex variables is, naturally, the
study of (differentiable) functions of more than one complex
variable. The rigid constraints imposed by complex differentiability
imply that, at least locally, these functions behave almost like
polynomials. In particular, study of the related spaces tends to
resemble algebraic geometry, except that tools of analysis are used in
addition to algebraic constructs. Differential equations on these
spaces and automorphisms of them provide useful connections with these
You might want to continue the tour with a trip through functional analysis.
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Last modified 1999/05/12 by Dave Rusin. Mail: email@example.com