Plan

Date	Monday	Tuesday	Thursday
2/01			Classes start. The complex numbers: algebraic properties, polar representation, arg z, Euler's formula, complex conjugate.
9/01	complex conjugate, relation to real/ imaginary parts and modulus, division, triangle and reverse triangle inequalities, sketching regions, powers and De Moivre's formula.	roots of unity, roots of complex nbs, regions (open, closed, deleted nbhd...) in the complex plane, functions of a complex variable.	Ex of functions of complex variables, principal root, mappings, z^2 in rectangular and polar coordinates, principal root.
16/01	Riemann Surface associated to the square roots, the exponential function, the inversion, def of limits and ex.	limit cont'd, continuity, properties of continuous functions, the Riemann Sphere.	The Riemann Sphere and the point at infinity, limit involving the point at infinity, differentiability.
23/01	Cauchy Riemann Equations, necessary and sufficient conditions, analytic functions.	ex of analytic and entire functions, singular points, examples, CR in polar coordinates, region of analyticity of the principal square root, results on analytic functions.	Harmonic functions, harmonic conjugates, elementary analytic functions: the exponential and the logarithm.
30/01	The branches of the logarithm, branch cuts, branch point, mapping of Log z, complex powers and roots, trigonometric functions.	Trigonometric functions cont'd, hyperbolic functions, complex integration: definite integral of a function of a real variable w(t), definition of an arc, parametrization, contour	Contour cont'd, contour integral, invariance of parametrization, example, path independence in contour integrals.
06/02	Proof of Cauchy's Thm, Cauchy-Goursat Thm, Simply-connected domains, extension of the Cauchy-Goursat thm to simply-connected domains, Thm on antiderivatives, ex	ex cont'd, estimates on the modulus of a contour integral, proof of the Thm on antiderivatives.	Test 1
13/02	Pf of antiderivative thm for continuous fcts in a domain cont'd (differentiability of the anti-derivative), Cauchy-Goursat for multiply connected domains, principle of path-deformation.	Cauchy Integral formula, applications and proof, integral formula for derivatives.	proof of the integral formula for derivatives, all derivatives of analytic functions are analytic.
Reading week
27/02	Ex of CIF for MC domains + formula for derivatives of analytic functions, pf that all derivatives of analytic functions are analytic, harmonic functions are infinitely differentiable, Morera's Thm, Cauchy's inequality, Liouville's thm.	Proof of Liouville's thm, Fundamental Thm of Algebra, Maximum Modulus Principle, pf part 1.	End of the pf of the Max mod principle, consequence for harmonic functions. Sequences, series, absolute convergence of series, power series.
06/03	power series cont'd: absolute and uniform convergence, radius of convergence, consequences of uniform convergence	uniform convergence of power series cont'd, consequences: power series are analytic, termwise integration and differentiation, Taylor series, radius of convergence of Taylor Series, uniqueness of representation, analytic functions are determined in their circle of convergence by their value and of all of its derivative at z_0	Many examples of Taylor Series
13/03	Proof of Taylor Series Thm, Laurent Series.	Snow day	More example of Laurent series.
20/03	Proof of Laurent Series, Residues, Cauchy's Residue Theorem.	Test 2	Cauchy's residue thm, example, pole of order m and removable singularities.
27/03	the three types of singularities, characterization of poles of order m and residue, zeros of analytic functions.	zeros of analytic functions are isolated, zeros and poles, another formula for residues around simple poles, more ex on finding residues.	Application of residue theory to improper integrals
03/04	More ex of improper integral: coming from Fourier series, and improper integrals with log	More on improper integrals, the Argument Principle	Argument Principle cont'd, Rouché's Thm.