Mathematical Physics 125B
Spring 2011
Lectures:
When:
Tuesday & Thursday,
9:30--10:50am. Where:
SSL
122
Discussions:
When:
Wednesday, 1:00--1:50pm.
Where: ICS
259
Instructor: Sasha
Chernyshev, Professor
Office: RH 310F
Phone: (949)-824-6440
E-mail: sasha@uci.edu
Office Hours:
Tuesday, 3:30--5:30pm
Required textbook
- M. L. Boas, Mathematical
methods in Physical Sciences, 3rd edition, Wiley, 2006.
[lower-lewel text]
Back-up and/or more challenging books
- J. Mathews and R. L. Walker, Mathematical methods of physics,
Addison-Wesley Publishing Cimpany, Inc., 1970.
- G. F. Carrier, M. Krook, and C. E. Pearson, Functions
of a complex variable: Theorie and Technique, McGraw-Hill book
company, New York, 1983. [an outstanding textbook on complex
analysis]
- G. B. Arfken, H. J. Weber, Mathematical methods for
physicists, Academic press, London, 2001. [a comprehensive
text on mathematical physics]
- G. Mahan, Applied Mathematics, Kluwer
Academic/Plenum Publishers, New York, 2002. [very practical book
on complex analysis and differential equations]
- James J. Kelly,
Graduate Mathematical Physics, With MATHEMATICA Supplements, Wiley-VCH,
2007
Extra material
- Asymptotic methods, (text).
- M&W, Appendix 1 (text).
M&W, Appendix 2 (text).
- CKP, bilinear mapping, etc. (text).
- CKP_I, Analytic functions, etc. (text).
CKP_II, Contour Integration (text)
- M&W, Chapter 1, WKB (text).
- M&W, Chapter 3 (integration and asymptotics) (text).
M&W, Chapter 5 (dispersion relations) (text).
- intro into Kronig-Pinney model (text).
Topics
In this course I will give an introduction into some of the
mathematical methods used in physics,
such as complex analysis, asymptotic methods, and differential
equations.
This is a rough list of topics
- Functions of a complex variable
- Complex numbers
- Analytic functions, conformal mapping
- Contour integration
- Asymptotic methods
- Asymptotic series
- Intergral methods (Laplace, steepest descent, stationary
phase)
- WKB
- Differential equations, Green's Functions
- Ordinary/partial differential equations
- Integral transforms (Fourier, Laplace)
- Green's Functions method
Here is a preliminary list
of topics in more detail
- Functions of a complex variable
- Complex numbers. Arithmetic operations. Polar representation.
de Moivre's formula.
- Elementary functions: power, exponent, trigonometric
functions, logarithm. Multiple-valuedness. Principal value.
- Inverse trigonometric/hyperbolic functions as logs
- Idea of conformal mapping. Complex functions as mappings:
exponent, power, fractional function.
- Stereographic projection. Fractional transformation.
- Branch points, branch cuts. Riemann surfaces. Examples.
- Differentiation in the complex plane. Complex derivative.
- Cauchy-Riemann equation conditions. Harmonic functions.
- Integration in the complex plane. Independence on contour.
Cauchy's theorem.
- Cauchy's integral formula. Cauchy's formulas for derivatives.
- Mean value theorem. Maximum modulus theorem. Cauchy's
inequalities and Liouville's theorem. Fundamental theorem of algebra.
- Taylor series. Analytic continuation. Laurent series. Ways of
deriving Laurent series.
- Types of isolated singularities: poles and essential
singularities. Singularities at infinity. Zeros and poles.
- Residue theorem. Contour integration. Residue calculus.
Examples of integrals.
- Singularities on the contour. Cauchy's principal value of the
integral.
- Residue calculus. Jordan's Lemma. Residue at infinity.
Examples of integrals: simple polynomials. Higher-order poles.
- Log-trick. Integrals of trigonometric functions.
- Avoiding/using/embracing branch-cuts. Contours by design.
- Integrals of special functions. Conversion of sums into
integrals.
- Asymptotic methods
- Asymptotic expansion, general idea. "Direct" methods:
integration by parts, expansion under the integral. Taylor vs.
asymptotic series.
- Laplace's method. Examples. Method of stationary phase.
Example. Method of steepest descent (saddle-point method).
- WKB equation and its solution from the saddle-point of the
Airy function.
- Asymptotic methods for differential equations. Stokes
phenomena.
- Differential equations and Integral transforms
- brief story of the DEs
- Fourier transform, reciprocal space, idea of mapping onto it
- Laplace transforms, why? Inverse LT.
- Inhomogeneous ODEs, solving with the FT or LT.
- Green's Function, the idea, whys and hows.
- Finding the GF. Integral transforms methods, using
homogeneous solutions+"stitching", role of BCs.
- Relation of the GF to the eigenvalue problem.
- Partial differential equations, more space, time variables,
equations in physics.
- 1D diffusion equation, heat pulse propagation (GF) by FT of
the space or LT of the time variables, discussion.
- Wave equation, wavefront propagation in 1D.
- BIG PICTURE
Assignments
There will be weekly homework assignments, midterm, and a final.
All assignments need to be turned in in hardcopy form. Some problems
will be specifically required to be solved with (or without!) Mathematica.
The grade for
the course will be based on: (i)
homework (35%)
(ii)
midterm
(30%)
(iii) final exam
(35%)