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 series and products of holomorphic functions, 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.
• Ahlfors L., Complex Analysis, McGraw-Hill, 1979
• 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, holomorphy 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 Solutions pdf



• October 12 : Removable, polar and essential singularities. Riemann extension theorem. The image of the functions exp and sin; examples of removable, polar and essential singularities. Laurent series, Cauchy internal integral (cf. Thm. 4.10); Cauchy external integral is similar. From Cauchy integral formula for an annulus to Laurent series. The principal part of a Laurent series is truncated (resp. vanishes) for a pole (resp. for a removable singularity).


• October 13: Four equivalent characterisations of a pole; three equivalent characterisations of an essential singularity; Casorati-Weierstrass’ theorem. Corollary: the preimage of every element in the image is necessarily infinite. Picard’s big theorem (no proof). The automorphism group of the complex plane C is the complex affine group. Polynomials are the only entire functions which extend holomorphically to the Riemann sphere. A third and more elegant proof of the fundamental theorem of algebra.


• October 14: Discrete subsets of singularities and residues. Cauchy’s integral formula revisited. Residue theorem. Computations of residues for polar singularities; general formula and explicit computation in the case of order one and two. Computations of real definite integrals by means of the residue theorem for certain classes of integrands, e.g. particular rational functions, the slit method and the sector method.


• October 16: The field of meromorphic functions on a domain D is the quotient field of the integral domain of holomorphic functions on D (hints to the proof). The argument principle (no proof) and its geometric interpretation. Rouché’s theorem. Corollaries: one more proof of the Fundamental Theorem of Algebra, localization of the zeros of a polynomial.


• Exercises 2 pdf Solutions pdf


Amine Zemirni


• November 12: Winding number concept; Cauchy's integral formula using Winding numbers; general form of the argument principle.


Laura Geatti


• November 16: various types of convergence in the space of holomorphic functions on a domain D⊂ C. Uniform convergence on compact sets and local uniform convergence. Weierstarss’ convergence theorem. Hurwitz’s theorem and some consequences: the uniform limit of a non-zero sequence of holomorphic functions is either identically vanishing or non-zero; the uniform limit of an injective sequence of holomorphic functions is either constant or injective. Montel’s theorem.


• November 17: The topology of uniform convergence on compact sets on the space of continuous functions C(D) on a domain D is induced by a metric d. The space (C(D),d) is a complete metric space and the subspace of holomorphic functions O(D) is a closed subspace. Characterization of compact sets in O(D).


• Exercises 3 pdf
  
  
• November 19: A proof of the Riemann mapping theorem.


• November 20: Discussion of Exercises 3. Infinite series of holomorphic functions: various types of convergence; normal convergence and the Weierstrass’ M-test. Example: the series expansion of the Riemann ζ -function in the region Re(z)>1. Introduction to infinite series of meromorphic functions: various notions of convergence, the sum of a locally uniformly convergent series is a meromorphic function (to be continued).


• November 23: Locally uniformly convergent series of meromorphic functions and their derivatives. An example: π² /(sin π z)² . Meromorphic doubly periodic functions on C: the Weierstrass ℘-function and its derivative. Mittag-Leffler theorem: existence of meromorphic functions on C with assigned set of poles and respective orders.


• November 24: Infinite products of complex numbers. Criteria of convergence. Infinite products of holomorphic functions. Construction of entire functions with assigned set of zeros and respective multiplicities. Weierstrass elementary factors. Weierstrass’ factorisation theorem. An example: sin π z. A meromorphic function on C is the quotient of entire functions.


• Exercises 4 pdf
  
  
• November 26: Criteria of convergence for an infinite product of Weierstrass elementary factors.


• November 27:


Amine Zemirni

• January ??: