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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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).

History

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

f(z) - The Complex Variables Program

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

Conformal mappings

Other web sites with this focus

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

Selected topics at this site

• We exclude elementary topics from this collection in general but the question "What is i^i?" is so frequently asked, it needs inclusion.
• Example of contour integration of definite integrals (here, of ln( a + sin(x) ) on [0, 2 pi])
• Bounds on the value of a polynomial
• Distributions of roots, and factorizations, of entire complex-analytic functions.
• What are meromorphic 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.
• Infinite product representations of functions from their zeros
• How does a Taylor series behave on the circle of convergence?
• The Taylor series for arcsin -- why is it so slow to converge?
• Connection between Taylor series and area of images.
• Löwner's Lemma (analytic function on the disc stretch circular arcs)
• Delicate estimates of Taylor series coefficients using contour integrals.
• No bounded analytic functions on the half-plane which vanish at integers, except zero
• How to find a conformal mapping between two complex domains?
• Conformal mappings to the interior of a curve or region between two curves.
• Proof of Weierstrass Approximation theorem using Bernstein polynomials
• Series of analytic functions with non-entire coefficients
• Entire functions with no 2-cycles (f(f(x))=x) nor fixed points are translations
• Citation to Quaternionic analysis.
• Definitions of analyticity do not generalize well to quaternions
• Comments about non-Archimedean fields.
• Introduction to Riemann surfaces (through elliptic integrals)
• Computing Riemann-surface functions g_2(L),g_3(L) from an integer lattice L
• Definite integrals of functions on Riemann surfaces over closed curves may be computable
• Reducing some non-elementary algebraic antiderivatives to elliptic integrals (Riemann surfaces)
• 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