nLab
complex analysis

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.

References

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

  • 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, 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

category: analysis

Revised on March 7, 2013 21:06:48 by Zoran Škoda (161.53.130.104)