- 10 Sequences
- Advanced Level
- Starts on April 22, 2019
- Ends on August 4, 2019
Complex Analysis with Physical Applications
You can't access an archived course
Course details
Syllabus
- Asymptotic series as approximation of definite integrals.
- Examples, optimal summation Taylor vs asymptotic expansions.
- Zero term and full Laplace asymptotic series.
- Asymptotics of Error and Fresnel integrals.
- 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.
- 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.
- Subtleties of a contour deformation.
- Contribution of end points.
- Higher order saddles.
- Coalescent saddle and pole.
- 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.
- 1D Coulomb potential
- Harmonic oscillator, method 1
- Restricted harmonic oscillator
- Harmonic oscillator, method 2
- 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.
- Solutions of the differential equations of higher order by Laplace method.
- More complicated examples.
- Killer problems
Prerequisite
Instructors
Yaroslav Rodionov
Associate Professor
National University of Science and Technology MISIS
Konstantin Tikhonov
Researcher, Theoretical Physics
Landau Institute
Platform
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