date_range Débute le 9 mai 2017
event_note Se termine le 7 mai 2019
list 14 séquences
assignment Niveau : Avancé
chat_bubble_outline Langue : Anglais
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Les infos clés

credit_card Formation gratuite
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timer 70 heures de cours

En résumé

In this advanced math course, you will learn how to build solutions to important differential equations in physics and their asymptotic expansions. Armed with the tools mastered in this course, you will have a solid command of the methods of tackling differential equations and integrals encountered in theoretical and applied physics and material science.

The course is for engineering and physics majors. The course instructors are active researchers in theoretical solid-state physics.

You will learn:

  1. Basics of asymptotic expansions.
  2. Special functions.
  3. Saddle point techniques.
  4. Laplace method of solving differential equations with linear coefficients.
  5. Stokes phenomenon.
  6. Quantum mechanical applications.

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Les prérequis

Good knowledge of real and basics of complex analysis, differential equations and general physics.

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Le programme

Week 1: Asymptotic series. Introduction.
1.1 Asymptotic series as approximation of definite integrals.
1.2 Taylor Series vs Asymptotic Expansions.
1.3 Optimal summation. Superasymptotics.
1.4 Taylor Series vs Asymptotic Expansions II (Illustration).
1.5 Integration by parts technique: limitations and more examples.
1.6 Estimation of reminder term.

Week 2: Laplace method and stationary phase approximation.
2.1 Laplace method: Introduction.
2.2 Laplace method: example.
2.3 Laplace method: Full asymptotic series.
2.4 Stationary phase approximation.

Week 3: Elementary special functions.
3.1 Euler’s Gamma function, definition and elementary properties.
3.2 Analytical continuation and examples of applications.
3.3 Stirling formula and its analytic continuation.
3.4 Computation of infinite products, examples.
3.6 Digamma function: properties and asymptotics.
3.7 Beta-function: definition, properties and examples..
3.8 Applications of digamma function.

Week 4: Saddle point approximation.
4.1 Saddle point approximation.
4.2 Application: relativistic particle in a corner.
4.3 Application: asymptotic of Legendre polynomials.
4.4 Application: Non-homogeneous exponent.

Week 5: Construction of solutions of DE by power series.
5.1 Representation of solutions of differential equations by convergent series.
5.2 Kummer's equation, full study.
5.3 Bessel Function, asymptotics.

Week 6: Physical Applications, I.
6.1 Bound state in 1D quantum mechanics.
6.2 Bound state in a shallow potential.

Week 7: Saddle point approximation II.
7.1 Saddle point approximation, end-points contribution.
7.3 Higher order saddles.
7.4 Coalescent saddle and pole.
7.5 Watson’s lemma.

Week 8: DE with linear coefficients.
8.1 Introduction into the method.
8.2 Examples: (building of exact solutions, choice of the contour, study of asymptotics, deformation of contours and branchcuts, normalization) .
a) Example 1; b) Example 2; c) Example 3; d) Example 4 (advanced)

Week 9: Physical applications, II.
9.1 1D Coulomb potential.
9.2 Harmonic oscillator 1.
9.3 Particle on a spring with a wall.
9.4 Harmonic oscillator 2(different ansatz, different contours).

Week 10. Stokes Phenomenon in asymptotic series and WKB.
10.1 Airy asymptotic series.
10.2 WKB.
10.3 Asymptotics of Airy's function in the complex plane.
10.4 Stokes Phenomenon.

Week 11. Differential EQS with linear coefficients, II.
11.1 Example1: equation of the third order, study of the structure of contours and asymptotics.
11.2 Example2 (advanced): equation of the third order, study of the structure of contours and asymptotics.

Week 12: Physical applications, III.
12.1 Over-barrier reflection, basic theory.
12.2 Over-barrier reflection, two turning points.
12.3 Advanced example: Over-barrier reflection from the turning point and the pole.

Week 13: Physical applications, IV.
13.1 Aharonov-Bohm effect, Introduction.
13.2 Partial wave decomposition (no flux).
13.3 Partial wave decomposition (with flux).
13.4 Asymptotic behavior and dislocations of the wave trains.

Week 14: Final Exam.
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Les intervenants

Yaroslav Rodionov
Associate Professor
National University of Science and Technology MISIS

Konstantin Tikhonov
Researcher, Theoretical Physics
Landau Institute

assistant

La plateforme

EdX est une plateforme d'apprentissage en ligne (dite FLOT ou MOOC). Elle héberge et met gratuitement à disposition des cours en ligne de niveau universitaire à travers le monde entier. Elle mène également des recherches sur l'apprentissage en ligne et la façon dont les utilisateurs utilisent celle-ci. Elle est à but non lucratif et la plateforme utilise un logiciel open source.

EdX a été fondée par le Massachusetts Institute of Technology et par l'université Harvard en mai 2012. En 2014, environ 50 écoles, associations et organisations internationales offrent ou projettent d'offrir des cours sur EdX. En juillet 2014, elle avait plus de 2,5 millions d'utilisateurs suivant plus de 200 cours en ligne.

Les deux universités américaines qui financent la plateforme ont investi 60 millions USD dans son développement. La plateforme France Université Numérique utilise la technologie openedX, supportée par Google.

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