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[Texts]## 30: Functions of a complex variable |

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).

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

- 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.

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.

Conformal package at Netlib

f(z) - The Complex Variables Program

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

- Here are the AMS and Goettingen resource pages for area 30.
- Some nice animations describing some complex maps.

- We exclude elementary topics from this collection in general but the question "What is i^i?" is so frequently asked, it needs inclusion.
- Distributions of roots, and factorizations, of entire complex-analytic functions.
- Functions in the convex hull of some functions of the form 1/(z-z_j) have roots in the convex hull of the z_j.
- Summaries of the Picard theorems.
- The Casorati-Weierstrass theorem.
- Connection between Taylor series and area of images.
- How does a Taylor series behave on the circle of convergence?
- Delicate estimates of Taylor series coefficients using contour integrals.
- How to find a conformal mapping between two complex domains?
- Conformal mappings to the interior of a curve or region between two curves.
- Citation to Quaternionic analysis.
- Comments about non-Archimedean fields.
- Recognizing complicated definite integrals as periods on a Riemann surface
- Conformal embeddings of Riemann surfaces into R^3.
- Global Complex Analysis is differential topology; low-dim manifolds which are groups

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org