complex analysis (changes) in nLab

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Complex analysis

Idea

Complex analysis is the mathematical analysis of complex-valued analytic (typically) functions of a complex variable, of several complex variables, or on a complex analytic manifold.

Since a complex number can be understood as a pair of real numbers, this would naively reduce to analysis of pairs of functions of an even number of real variables; however by complex analysis we mean mathematical analysis which takes into account limits and derivative which do not depend on the real line in a complex plane on which we approach a point. This leads to the notions of holomorphic function, meromorphic function, etc. which are the main subject of complex analysis. However, there are connections to such real-analytic notions as harmonic analysis, and the geometric approach to complex analysis builds on the theory of smooth functions.

• Weierstrass preparation theorem, several complex variables, spectral theory, Riemann surface

References

• Wikipedia (English): Complex analysis, Several complex variables

• John B. Conway, John B. Conway?, Functions of one complex variable, Springer 1978; Functions of one complex variable II, GTM 159

• Lars V. Ahlfors, Complex analysis, McGraw-Hill, 1966.

• Raghavan Narasimhan, Complex analysis in one variable, Birkhäuser, 1985.

• Lars Hörmander, An introduction to complex analysis in several variables, North-Holland

• Steven G. Krantz, Geometric function theory: explorations in complex analysis, Birkhäuser

• Robert C. Gunning, Hugo Rossi, Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing

• Douglas N. Arnold, Complex analysis, lecture notes, pdf

• Elias M. Stein, Rami Shakarchi, Complex analysis, Princeton University Press 2003, 2012

Via constructive analysis:

• Errett Bishop, Chapter 5 of: Foundations of Constructive Analysis, McGraw-Hill (1967)

• Errett Bishop, Douglas Bridges, Chapter 5 of: Constructive Analysis, Grundlehren der mathematischen Wissenschaften 279, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]