Analysis IV (for SV, MT)

Examen mercredi 18 juin 9h15-12h15.

Le cours aura lieu les lundis entre 15h15 et 17h à la salle CO 1 près de L'Esplanade.

Les séances d'exercices auront lieu toute de suite après, entre 17h15 et 19h, aux salles CE1100, CE1103, CM1104, CM1105, CM1221

Nous allons étudier les chapitres 9-12 et 16-18 du livre de Dacorogna et Tanteri.

Lecture 2: Recollection on integration of vector fields and Green's Theorem (chapter 3 and 4). Integration of a complex function along a curve, Cauchy's Theorem, Cauchy's integral formula (Chapter 10).

Lecture 4: More Examples of Laurent expansion: poles, essential singularities and ray of convergence (Chapter 11). Computation of Residues, Theorem of Residues (Chapter 12) and examples.

Exercises: I suggest looking at Exercise 11.7,11.9,11.10,11.11,11.12 and 11.15.

Lecture 5: Residue Theorem (Chapter 12).

Lecture 6: With this Lecture we conclude the Complex Analysis part of the course. For the exam, the following competences are necessary: 1) Be able to determine weather a given function is holomorphic in a given domain and be able to compute the complex derivative (Chaper 9) 2) Understand how the integration along a curve of a complex function is defined; be able to compute complex integrals using either the definition (for simple one) or Cauchy Theorem (and its Corollaries) and Cauchy Integral Formula. (Chapter 10) 3) Be able to compute the Taylor/Laurent expansion of a complex function satisfying the hypothesis of Theorem 11.1 and to determine the order of pole at a point z_0 of a given complex function; be able to compute the Residue at a given point of a given complex function (Chapter 11). 4) Be able to apply the Theorem of Residues (in particular understand how to control that its hypothesis are verify) to compute complex or real integrals; for real integrals on the real line, be able to prove the convergence to 0 of the auxiliary complex integral along the upper semi-circle ( Chaper 12)

Lecture 7: Reminder on Furier transform (Chapter 15). Definition and properties of the Laplace transform (Chapter 16) plus Examples. 

Reminder of: Definition and properties of Laplace transform (Chapter 16); application to ordinary differential equations ( 1st order; the mass-spring-dumper example.) 

Inversion Formula for Laplace transform (Theorem 16.3); a detailed Example: Example 16.7.

Other Exercises and Examples to look at: Exercise 16.4 (sketch of solution in the notes); Exercise 16.7; Example 17.1.

Lecture 9: Reminder of Laplace inverse transform formula; solving ODE using Laplace transform

Lecture 10. Example of PDE (chapter 18); introduction to distributions: the Dirac mass example; definition in general; Examples.

Lecture 11: More on distributions: Derivatives, Laplace transform and convolution. Differential equations for distributions. Fundamental solution of the wave equation. 

Lecture 12: Distributions and fundamental solution for PDE. Fouries series, solution to heat equation on the interval using Fourier series. (Chapter 14+Chapter 18)

Lecture 13: recap on how to solve Heat and Wave equation on an interval using Fourier series. Recollection of properties of the Fourier transform and application to solution of Heat and Wave equations over the all real line.

What you need to know on distributions: definition of test function and distribution 

(Gerrit, section 2.2. Warning: he defines distributions as linear forms from the set of test functions to complex numbers instead of real numbers; a part from this, it is the same definition we saw in class!)

Distributional derivative; what does it mean to solve a distributional partial differential equation  (Section 3.1; Examples 1 and 2 and 3 of Section 3.2. Also look at Exercises 3.2 and 3.3)

 Laplace transform of a distribution which is defined as the derivative of a PW continues function.

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