Department
of Physics and Astronomy
PHYS 6520/4520 (same as Math 6265/4265)
MATHEMATICS OF PHYSICS
II
SUMMER SEMESTER 2001
Tuesday, Thursday 4:45 pm - 7:30
pm --- 527 General Classroom Building
Instructor: Mark Stockman
Office: 455 Science Annex
Phone: (404)651-2779
E-mail: mstockman@gsu.edu
Internet: http://www.phy-astr.gsu.edu/stockman/
Grading: Homework 40%, midterm exam 20%, final exam 40%.
Final Exam: Friday, August 2 from 5 to 7 pm in
the regular classroom (527 G)
Text: Erwin Kreysig, Advanced Engineering Mathematics (Eighth
Edition), J. Wiley, New York, 1999
Prerequisite: Math 3260
SYLLABUS
1. Ordinary Differential Equations and Special Functions (Chapter
4)
-
Power series solutions
-
Legendre equation and polynomials
-
Frobenius method
-
Bessel equation and functions
-
Sturm-Liouville problem and orthogonal functions
-
Solutions by orthogonal eigenfunction expansions
2. Fourier Series and Integrals (Chapter 10)
-
Periodic functions
-
Fourier series
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Even functions and Cosine series; Odd functions and Sine series
-
Complex Fourier Series
-
Fast Fourier transform (FFT)
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Example: Solving periodic and aperiodic mechanical problems using FFT
3. Partial Differential Equations (Chapter 11 with
some material of Chapter 4)
-
Introduction. Equations of mathematical physics: Hyperbolic (wave equation),
parabolic (diffusion or heat equation and Schroedinger equation), and elliptic
(Laplace equation and Poisson equation)
-
Solving one-dimensional wave equation (vibrating spring): Separation of
variables and Fourier expansion
-
General solutions of differential equations: D'Alembert solution of the
wave equation
-
Diffusion equation: Solution by Fourier series and integrals.
-
Two dimensional wave equation: Vibrating rectangular membrane
-
Laplacian in polar coordinates. Use of Fourier-Bessel series. Vibrating
circular membrane
-
Laplace equation in three dimensions. Spherical Bessel functions, Legendre
polynomials and functions, and spherical harmonics. Potential problems
-
Green's functions
4. Analytic Functions (Chapter 12) [Time
permitting]
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Complex numbers, complex plane, modulus and argument
-
Derivative, analytic functions
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Cauchy-Riemann and Laplace equations
-
Conformal mapping
-
Analytic properties of elementary functions: square root, exponential function,
trigonometric and hyperbolic functions, and logarithm
-
Branch cuts and Riemann surfaces
5. Complex integration (Chapter 13) [Time
permitting]
Complex line integral
Cauchy theorem and integral
Derivatives of analytic functions
6. Residue integration (Chapter 15) [Time
permitting]
-
Laurent series
-
Poles and residues
-
Residue integration method
-
Evaluation of real integrals