• Length:
    10 Weeks
  • Effort:
    3–5 hours per week
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  • Course Type:
    Instructor-led on a course schedule


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

About this course

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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.

What you'll learn

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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.
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

Meet your instructors

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

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Learner testimonials

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

Who can take this course?

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