An Introduction to Riemann Surfaces pp 3-23 | Cite as

# Complex Analysis in ℂ

## Abstract

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. 1.3–1.6.

## Keywords

Riemann Surface Compact Riemann Surface Exterior Derivative Weierstrass Theorem Riemann Equation## References

- [Hö]L. Hörmander,
*An Introduction to Complex Analysis in Several Variables*, third edition, North-Holland, Amsterdam, 1990. MATHGoogle Scholar - [Ns5]R. Narasimhan,
*Complex Analysis in One Variable*, second ed., Birkhäuser, Boston, 2001. MATHCrossRefGoogle Scholar