• Duración:
    10 semanas
  • Dedicación:
    3–5 horas por semana
  • Precio:

    GRATIS
    Agregar un Certificado Verificado por $49 USD

  • Institución
  • Tema:
  • Nivel:
    Advanced
  • Idioma:
    English
  • Transcripción de video:
    English
  • Tipo de curso:
    Al ritmo del instructor

Prerrequisitos

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

Sobre este curso

Omitir Sobre este curso

The course is for engineering and physics majors.
You will learn how to build the solutions of important in physics differential equations and their asymptotic expansions.

The main topics include:
1.    Introduction to asymptotic series.
2.    Special functions.
3.    Saddle point techniques.
4.    Laplace method of solving differential equations with linear coefficients.
5.    Stokes phenomenon.

The course instructors are active researchers in a theoretical solid state physics.  Armed with the tools mastered while attending the course, the students will have solid command of the methods of tackling differential equations and integrals encountered in theoretical and applied  physics and material science.

Lo que aprenderás

Omitir Lo que aprenderás

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.

Plan de estudios

Omitir Plan de estudios
Week 1. Asymptotic series. Introduction 
  • Asymptotic series as approximation of definite integrals. 
  • Examples, optimal summation Taylor vs asymptotic expansions.
Week 2. Laplace-type integrals and stationary phase approximations
  • Zero term and full Laplace asymptotic series.
  • Asymptotics of Error and Fresnel integrals.
Week 3. Euler Gamma and Beta-functions, analytic continuation and asymptotics
  • Euler Gamma function: definition, functional equation and analytic continuation.
  • Hankel representation for Gamma-function. 
  • Beta and digamma functions.
  • Asymptotic expansions.
  • Application of Gamma functions for the computation of integrals.
Week 4. Saddle point approximation I 
  • Introduction to the method of saddle point approximation.
  • The search for optimal deformation of the contour.
  • Full asymptotic series.
  • Elementary applications of the saddle point approximation.
Week 5. Saddle point approximation II
  • Subtleties of a contour deformation.
  • Contribution of end points.
  • Higher order saddles.
  • Coalescent saddle and pole.
Week 6. Differential equations with linear coefficients. Laplace method I
  • Construction of the solution of the differential equations with linear coefficients in terms of Laplace type contour integrals.
  • Examples of solutions of second order differential equations
  • The general outline of the technique.
Week 7. Physical applications
  • 1D Coulomb potential  
  • Harmonic oscillator, method 1
  • Restricted harmonic oscillator 
  • Harmonic oscillator, method 2
Week 8. Stokes Phenomenon in asymptotic series and WKB approximation in Quantum Mechanics
  • Solution of Airy's equation by asymptotic series.
  • WKB approximation for solution of wave equations.
  • Asymptotics of Airy's function in the complex plane.
  • Stokes phenomenon.
Week 9. Differential equations with linear coefficients. Laplace method II (higher order equations)
  • Solutions of the differential equations of higher order by Laplace method.
  • More complicated examples.
  • Killer problems
 Week 10. Final Exam

Conoce a tus instructores

Yaroslav Rodionov
Associate Professor
National University of Science and Technology MISIS
Konstantin Tikhonov
Researcher, Theoretical Physics
Landau Institute

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Testimonios de los estudiantes

Useful but challenging course. Quite helpful for future progress in quantum mechanics.

¿Quién puede hacer este curso?

Lamentablemente, las personas de uno o más de los siguientes países o regiones no podrán registrarse para este curso: Irán, Cuba y la región de Crimea en Ucrania. Si bien edX consiguió licencias de la Oficina de Control de Activos Extranjeros de los EE. UU. (U.S. Office of Foreign Assets Control, OFAC) para ofrecer nuestros cursos a personas en estos países y regiones, las licencias que hemos recibido no son lo suficientemente amplias como para permitirnos dictar este curso en todas las ubicaciones. edX lamenta profundamente que las sanciones estadounidenses impidan que ofrezcamos todos nuestros cursos a cualquier persona, sin importar dónde viva.