Complex Analysis in ℂ

Part of the Cornerstones book series (COR)


In this chapter, following Hörmander (An Introduction to Complex Analysis in Several Variables, North-Holland, 1990), we consider elementary definitions and facts concerning complex analysis in ℂ from the point of view of local solutions of the inhomogeneous Cauchy–Riemann equation \(\partial u/\partial\bar{z}=v\). The global solution of the analogous inhomogeneous Cauchy–Riemann equation on a Riemann surface (see Chaps.  2 and  3) will allow us to obtain analogues of some of the central theorems of complex analysis in the plane (for example, the Riemann mapping theorem, the Mittag-Leffler theorem, and the Weierstrass theorem) for open Riemann surfaces, as well as some of the central theorems of the theory of compact Riemann surfaces (for example, the Riemann–Roch theorem). A reader who is familiar with complex analysis in the plane may wish to read Sects. 1.1 and 1.2 carefully, but only skim Sects. 1.31.6.


Riemann Surface Compact Riemann Surface Exterior Derivative Weierstrass Theorem Riemann Equation 
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  1. [Hö]
    L. Hörmander, An Introduction to Complex Analysis in Several Variables, third edition, North-Holland, Amsterdam, 1990. MATHGoogle Scholar
  2. [Ns5]
    R. Narasimhan, Complex Analysis in One Variable, second ed., Birkhäuser, Boston, 2001. MATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2.Department of MathematicsSUNY at BuffaloBuffaloUSA

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