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.
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.
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.
Week 9. Differential equations with linear coefficients. Laplace method II (higher order equations)
Solutions of the differential equations of higher order by Laplace method.