Mathematical Physics 212A
Fall 2007

Lectures: When: Tuesday & Thursday, 12:30--1:50pm. Where: DBH 1425 (Donald Bren Hall, entrance from near the faculty club)
Discussions: When: Wednesday, 11:00--11:50am. Where: MSTB 110 (PC Lab)

Instructor: Sasha Chernyshev, Associate Professor
Office: FRH 2158
Phone: (949)-824-6440
E-mail: sasha@uci.edu
Office Hours: Thursday, 2--3:30pm

Required textbook

J. Mathews and R. L. Walker, Mathematical methods of physics, Addison-Wesley Publishing Cimpany, Inc., 1970.

Required software

Mathematica (Student's edition)

Recommended E-book: MathematicaHandbook,by Prof. Peter Taborek, www.MathematicaHandbook.com

Back-up and/or more challenging books

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]

M. L. Boas, Mathematical methods in Physical Sciences, 3rd edition, Wiley, 2006. [lower-lewel text]

Extra material

Asymptotic methods, (text).

Green (text).

Topics

In this course I will give (i) an introduction into some of the mathematical methods used in physics,
such as complex analysis, asymptotic methods, and differential equations, and
(ii) an introduction into the basics of Mathematica software that can be used for a wide variety of problems.

This is a rough list of topics

Introduction to Mathematica
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 more detailed/realistic list of topics to be covered in the course.

Mathematica intro
- General idea, syntax, Help, arithmetics, elementary functions, symbolic expressions, manipulating them [Expand, etc.]
- assigning values, user-defined functions, calculus [D, Sum, etc.], plotting [Plot, Show]
- lists, matrices, defining [Table], operations with them
- solving algebraic equations [Solve], numerical solution of the non-algebraic equations [FindRoot]
- parametric, 3D plots, controls [color], etc.
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%)