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Complex variables are often accepted in other parts of analysis when this causes no essential change in the theory; but here we focus on 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. 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 charges.
• 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 other areas.

You might want to continue the tour with a trip through the Theory of Functions.