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# 30: Functions of a complex variable

## Introduction

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).

## Applications and related fields

Problems involving complex numbers, rather than functions, are likely to be topics in algebra; see especially 12: Fields.

For analysis on manifolds, See 58-XX

Specific functions (e.g. the Gamma function) are treated with special functions or, in the case of the zeta function and its relatives, with analytic number theory

## Subfields

• 30A: General properties
• 30B: Series expansions
• 30C: Geometric function theory
• 30D: Entire and meromorphic functions, and related topics
• 30E: Miscellaneous topics of analysis in the complex domain
• 30F: Riemann surfaces
• 30G: Generalized function theory
• 30H05: Spaces and algebras of analytic functions. See also 32E25, 46EXX, 46J15

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Hamilton, Hugh J.: "A primer of complex variables, with an introduction to advanced techniques", Wadsworth Publishing Co., Inc. Brooks/Cole Publishing Co., Belmont, Calif. 1966 227 pp. MR34#1489

Beardon, A. F.: "A primer on Riemann surfaces", London Mathematical Society Lecture Note Series, 78; Cambridge University Press, Cambridge-New York, 1984. 188 pp. ISBN 0-521-27104-5 MR87h:30090

Sario, L.; Nakai, M.: "Classification theory of Riemann surfaces", Die Grundlehren der mathematischen Wissenschaften, Band 164; Springer-Verlag, New York-Berlin 1970 446 pp. MR41#8660

Bers, Lipman: "What is a Kleinian group? A crash course on Kleinian groups", (Lectures Special Session, Annual Winter Meeting, Amer. Math. Soc., San Francisco, Calif., 1974), pp. 1--14. Lecture Notes in Math., Vol. 400, Springer, Berlin, 1974. MR52#14277

"Reviews in Complex Analysis 1980-1986" (four volumes), American Mathematical Society, Providence, RI, 1989. 3064 pp., ISBN 0-8218-0127-9: Reviews reprinted from Mathematical Reviews published during 1980--1986.

## Software and tables

Conformal package at Netlib

There is a collection of programs for personal computers at the Mathematics Archives