Elementary functions of one complex variable: polynomials, exponential, logarithmic and trigonometric functions, their inverses. Real and complex differentiability. Holomorphic functions. Conformal mappings.
Piecewise smooth and rectifiable paths and curves. Complex integration along curves. 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 holomoprhic functions. Mean value property. Maximum modulus principle. Radius of convergence of power series. Cauchy-Hadamard formula. Theorem of Morera. Cauchy's estimates. Liouville's theorem. 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. Casorati-Weierstrass theorem.
Residues, their caculation. Residue theorem.
The argument principle. Rouché's theorem. Open mapping theorem. The univalence theorem (local injectivity criterion). Inverse function theorem. Branched covering structure theorem.
Spaces of holomorphic functions. Seminorms. Montel's theorem. Biholomphic maps between open sets. The Riemann mapping theorem.
Schwarz Lemma. Automorphisms of the disk. Homogeneity of the disk. Cayley transform. Automorphisms of the upper half-plane. Möbius transformations. Riemann sphere. Holomorphic and meromorphic functions on the Riemann sphere. Automorphisms of the Riemann sphere.
Schwarz Reflection Principle.
For exam-related problems look in TCD past examination papers and Mathematics department examination papers.
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