On successful completion of this module students will be able to;
Operate with holomorphic functions and branches of multi-valued holomorphic functions.
Give the appropriate definitions, statement and proofs of Cauchy theorem and its consequences.
Demonstrate the use of Morera and Rieman Extension theorems.
Give examples of power and Laurent series and of islolaated singularities that are removable, poles and essential.
Real and complex differentiability. Holomorphic functions. Branches of multi-valued functions. Branchs of logarithm and of the nth root. Conformal mappings.
Complex integration along piecewise smooth paths. Antiderivatives. Calculating integrals using antiderivatives. Cauchy's theorem: Goursat's version for a triangle, for star-shaped regions and their unions, homotopy version. Elements of homology and homological version of Cauchy's theorem.
Cauchy's integral formula. Power series expansion of holomorphic functions. Mean value property. Maximum modulus principle. Radius and disk of convergence of power series. Cauchy-Hadamard formula. Theorem of Morera, Cauchy's estimates. Liouville's theorem. Application to the Fundamental Theorem of Algebra. Compact convergence and Weierstrass theorem.
Order of zeroes. The identity principle. Laurent series expansion in a ring. Isolated singularities. Removable singularities, poles, essential singularities. Riemann extension theorem. Meromorphic functions. Casorate-Weierstrass theorem.
The argument principle. Rouché's theorem. Open mapping theorem. The univalence theorem (local injectivity criterion). Inverse function theorem.
Spaces of holomorphic functions. Seminorms. Montel's theorem. Biholomphic maps between open sets. The Riemann mapping theorem.
Möbius transformations. Riemann sphere (extended complex plane). Stereographic projection. Rationality of meromorphic functions on the Riemann sphere. Automorphisms fo the Riemann sphere an dthe complex plane. Schwarz Lemma. Automorphisms of the disk. Cayley transform. Automorphisms of the upper half-plane. Homogeneity of the Riemann sphere, complex plane and disk.