International Mathematics Master 2025-26
Algeria
COMPLEX ANALYSIS
Laura Geatti, Andrea Iannuzzi, Mohamed Amine Zemirni
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Program: This course discusses fundamental topics in the function theory of one complex variable:
basic properties of holomorphic functions,
local behaviour near a singularity, approximation theorems, infinite products, Riemann mapping theorem, entire functions and meromorphic functions.
Prerequisites: An introductory course in complex analysis.
- Zemirni M.A., Complex Analysis, Notes 2024/2025 pdf.
- Exercises 0 pdf
References:
- Sarason D., Notes on complex function theory, A.M.S. 2007.
- Cartan H., Elementary theory of analytic functions of one and several variables, Dover Public. Inc.,
1995.
- Greene R., Krantz S., Function theory of one complex variable, Pure
and Applied Math., John Wiley & Sons, 1997.
- Zemirni M.A., Entire functions theory, Notes 2024/2025 pdf.
- Bergweiler W., Entire and meromorphic functions, Notes 2017/2018.
Lectures overview:
Andrea Iannuzzi
- October 5: Mainly notation and review of the prerequisites: C-differentiable and holomorphic functions, CR equation and “ the complex differential operators” ∂z, ∂ z, the usual derivation rules, holomorphicity vs. C-linearity of the differential, inverse function theorem for holomorphic functions; the inverse is holomorphic! (more in the next future). Complex integration, existence of holomorphic primitive (Zemirni’s notes Cor. 4.5), Cauchy-Goursat theorem (Thm. 4.6), examples and an important application: existence of harmonic conjugates on a simply connected domain. Cauchy’s integral formula on simply connected domains (Thm. 4.8) and its consequences: Gauss mean value Theorem (Cor. 4.9), integral formula for the derivatives (Thm. 4.12), Cauchy’s inequalities (Cor. 4.13), Liouville’s theorem (Thm. 4.14), a different proof of the Fundamental Theorem of Algebra (cf. Thm. 2.2), polynomials extend continuously to the Riemann sphere P1( C) (much more than continuous, as we shall see later).
- October 6: Uniqueness of analytic continuation (Analytic Continuation Principle) in various versions. Examples/applications. (Be aware when extending a holomorphic function to different domains with disconnected intersection).
ACP for harmonic functions using harmonic conjugates on a smaller simply connected domain. What about real analytic functions? The more topologically oriented identity principle; the simple and intuitive case of polynomials. Multiplicity of a holomorphic function at a point.
The identity principle follows from the factorisation theorem (it will proved in the next lecture).
- October 7: Proof of the factorisation theorem. A subtle corollary: a non constant holomorphic map is locally a power of a local biholomorphism. Consequences: the open mapping theorem and
local injectivity imply the non vanishing of the complex derivative. The open mapping theorem for harmonic functions (exercise). The maximum modulus principle and its formulation on relatively compact subdomains. MinMax principle for harmonic functions. Phragmen-Lindelof theorem for sectors (no proof).
- October 8: Schwarz’s lemma. A simple trick to compute automorphism groups. The automorphism group of the unit disc. Two different realisations: Moebius transformations, SU(1,1)/ ± I2. The Cayley transform from the unit disc to the upper half plane H+. Another realisation of the automorphisms of the disk Aut(Δ)≅ Aut(H+), namely SL(2,R)/± I2.
The Riemann sphere P1( C) is a Riemann surface; its simplest atlas. Computation of Aut(P1( C)) assuming we
know that Aut(C) consists of affine transformations (we will prove it soon). Riemann uniformisation theorem (no proof, of course). One calls a Riemann surface elliptic, hyperbolic or parabolic according to whether its universal covering is isomorphic to P1( C), Δ or C.
- Exercises 1 pdf
- October 12:
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- October 15:
- October 16:
Amine Zemirni
Laura Geatti
- November 16:
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- November 27:
Amine Zemirni