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.
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 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.
10.3 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.
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