Coopération Mathématique Interuniversitaire Cambodge France,
Master Training Program
Complex Analysis
Phnom Penh, April 21 - May 9, 2008
Syllabus:
Holomorphic functions, Cauchy-Riemann equations, harmonic functions.Examples: polynomials, the exponential and trigonometric functions, complex logarithm.
Power series, analytic functions.
Integration on paths, Cauchy's theory.
Singularities, residues. Meromorphic functions.
Series and products of analytic or meromorphic functions.
Bibliographic references:
A number of basic courses on complex analysis and analytic functions of one complex variable are available the internet including the two next items:
Complex Variables by Robert B. Ash and W.P. Novinger
and
Analytic functions by Stanisław Saks and Antoni Zygmund, Monografie Matematyczne Tom 28, Warszawa-Wrocław 1952.
Further reference: S. Lang, Complex Analysis.
Course given in April/Mai, 2008:
Course 1, Monday, April 21: Introduction to complex numbers, irreducible polynomials over the real or the complex field, geometry of the complex plane. Complex functions: polynomials, rational functions, the exponential function.Course 2, Tuesday, April 22: The exponential function, the logarithm as a series in the disc of center 1 radius 1. Some differential equations. Solving z^n=u, e^z=1, z^2=1+i, z^4=1-i.
Course 3, Wednesday, April 23: Real functions: C^k, C^infinity, power series, differentiable functions of two variables. Complex functions, complex derivability, Cauchy-Riemann equations. Notation dz and dz bar, partial derivatives with respect to z and a bar. Power series, radius of convergence.
Course 4, Thursday, April 24: The differential of a function as a linear map. Connected open subsets of the complex plane, arcwise connectivity. The algebra of convergent power series in a disc. Units in this algebra: computing the inverse series.
Course 5, Friday, April 25: Composition of series. Binomial series, definition of (1+z)^w for z and w complex numbers with |z|<1. Checking relations like (1+z)^w=exp(w log(1+z)). Order of a power series. Order of zero of a power series expansion. The C-algebra H(U) of functions having a power series expansion at each point of U.
First short test, April 25
Course 6, Monday, April 28: Back to the correction of the first test. Examples of elements in H(U). Properties of these elements. Order of multiplicity of a zero. Local isomorphisms. Open mapping theorem.
Course 7, Tuesday, April 29: Study of 1/f(z) for f in H(U). Bijective maps, inverse. Exponential and logarithm.
Course 8, Wednesday, April 30: Complex integrals, properties, examples. Change of parameters. Length of a curve. Bound for an integral. Primitives of continuous functions. Goursat's theorem for a rectangle.
Course 9, Friday, May 2: Solution of exercises 17 and 18 of the first sheet of exercises. Proof of Goursat's theorem for a rectangle and for a triangle.
Exercises, May 2 and LaTeX source file.
Second short test, May 2
Course 10, Monday, May 5: Correction of the second test. Goursat's Theorem for a rectangle, for a triangle. Starlike open sets. Existence of a primitive for an analytic function. Cauchy's Theorem for a disc.
Course 11, Tuesday, May 6: Cauchy's integral formula for a disc. Holomorphic and analytic functions are the same. Isolated zeroes, unicity of analytic continuation. Cauchy's inequalities. Liouville's Theorem. Proof of D'Alembert's Theorem.
Course 12, Wednesday, May 7: Maximum modulus principle. Schwarz Lemma. Analytic functions in an annulus: sum of an analytic function in a disc and an analytic function outside a disc. Examples.
Course 13, Thursday, May 8: Laurent's development of an analytic function in an annulus. Isolated singularities of an analytic function. Residues. Residue theorem for a compact with an oriented boundary. Examples of computation of integrals.
Course 14, Friday, May 9: Exponential and logarithm.
Main test, May 9 and LaTeX source file.