Mathematical Physics 212A
Fall 2018

Lectures:

When: Tuesday & Thursday, 9:30am-10:50am. Where: RH 192 [Rowland Hall, Bldg. #400]

Discussions:

When: Wednesday, 11:00--11:50am. Where: PSCB 230

Instructor: Sasha Chernyshev, Professor
Office: RH 310F
Phone: (949)-824-6440
E-mail: sasha@uci.edu
Office Hours: Tuesday, 1:00pm--4:30pm or by appointment

Recommended textbooks (relevant chapters are posted)

James J. Kelly, Graduate Mathematical Physics, Wiley, 2006. [a "re-issue" of Mathews and Walker (below) with some extensions and Mathematica examples]

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

G. Mahan, Applied Mathematics, Kluwer Academic/Plenum Publishers, New York, 2002. [very practical book on complex analysis and differential equations]

Recommended software

Mathematica (Student's edition)

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

Back-up, follow-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, and F. E. Harris, Mathematical methods for physicists, Seventh Edition: A Comprehensive Guide, Academic press, 2012. [a comprehensive text on mathematical physics]

Extra material

Green (text).
Bayes on Gamma-function (text).

Useful Mathematica program to calculate integrals over a closed circle (center =c, radius =r) of f(z)

c_int.nb

Topics

This course is on some of the mathematical methods used in physics,
such as complex analysis, asymptotic methods, and differential equations.

Here is a brief and rough list of topics

Functions of a complex variable
- Complex numbers
- Analytic functions, conformal mapping
- Contour integration
Asymptotic methods
- Intergral methods (Laplace, steepest descent, stationary phase)
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. Links to relevant chapters of relevant texts are provided.

Functions of a complex variable [Kelly 1], [Kelly 2], [MW, concise intro into complex variables], [MW, mixed topics], [CKP1], [CKP2]
- 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. [CKP]
- Stereographic projection. Fractional transformation.
- Branch points, branch cuts. Riemann surfaces. Examples.
- Differentiation in the complex plane. Complex derivative. [CKP1]
- 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. [MW,Ch. 3, CKP2, Kelly 2, Mahan]
- Integrals of special functions. Conversion of sums into integrals.
Asymptotic methods [Mahan] [Kelly], [MW, Ch. 3, end] [Gamma function expansion via MW method by Mathematica]
- Method of steepest descent (saddle-point method). [Mahan]
Differential equations and Integral transforms [Mahan, DEs], [Mahan, transforms], [MW, DE 1], [MW, DE 2], [MW, transforms, Ch. 4], [Kelly, transforms], [Kelly, BV problem]
- 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 8 homework assignments, a midterm, and a final.
Homework assignments will be discussed and solved at the discussion sections.
Every student in this class will be asked to present a solution of one or two problems at one of those discussions.
The homework grade will be based on that presentation. Further participation in the discussions will be awarded extra credit.
The use of Mathematica is allowed for numerical verifications as well as for hints.
Analytical results should be obtained analytically.
The grade for the course will be based on:
(i) homework (20%)
(ii) midterm (30%)
(iii) final exam (50%)