Mathematical Physics 125A
Fall 2006

Lectures: Monday, Wednesday, Friday, 9:00-9:50am
MSTB 114

Discussion: Wednesday, 11:00-11:50am, PSCB 240

Introduction

Topics to be covered

• Special functions
  • General idea, common features
  • factorial, Gamma function, integral form, real, negative, singular points
  • Recursion relations, half-integer argument
    
  • Beta-function, relation to Gamma, pendulum problem
  • Elliptic integrals, Error function
  • Asymptotic expansion, general idea, erfc(x) example
  • convergent v.s. asymptotic
  • Laplace (saddle-point) method, derivation of the Stirling's formula for Gamma(x)
  • Review of DE classifications, series solution
  • "trivial" methods: integrating factor (1st order), finding the "second" solution (2nd order)
  • Legendre equation, series solution, Legendre polynomials
  • Generating function, recursion relations, Rodrigues' formula
  • Orthogonality, normalization, completeness
  • Asociated Legendre functions, spherical harmonics
  • Method of Frobenius, Bessel's equation, functions, the second solution
  • Generating function (integer order), integral representation
  • orthogonality, normalization, the other Bessel's functions
  • asymptotics of the Bessel's functions, stationary phase method
    
  • asymptotic methods for differential equations, Bessel, Hermite, and Laguerre equations
  • Hermite equation, ladder operators
    
• Differential equations and Integral transforms
  • loose ends, brief story of the non-linear DEs, 1st order, 2nd, quadrature method
  • Fourier series, completeness, symmetry, overshot for the step-finctions (Gibbs phenomenon), relation to the asymptotic expansion
  • reciprocal space, idea of mapping onto it
  • continuous+finite <-> infinite+discrete, discrete+finite <-> finite+discrete, and continuous+infinite <-> infinite+continuous
  • Fourier transforms, density transforms. sin, cos-versions of FT, higher dimensions, gaussian
  • Laplace transforms, why? Inverse LT. Tables of LTs (and FTs), derivatives, multiplications, shifts, etc.
  • Convolution, its applications.
    
  • Inhomogeneous ODEs, solving with the FT or LT.
  • Green's Function, the idea, whys and hows.
  • delta-function, properties, representations; limits of functions, integral forms.
  • relation to step-function, use for the GF method.
  • Finding the GF. Integral transforms methods, using homogeneous solutions+"stitching", role of BCs
  • Examples, 1D Laplace equation, Fourier form of the solution. Relation of the GF to the eigenvalue problem
  • GF representation in terms of the eigenfunctions and eigenvalues
  • 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
    
  • Separation of variables method, separation constant, 2D Laplace, example.
  • Wave equation, notes (eigenmodes) of the string, rectangular drum, and circular drum
  • wavefront propagation in 1D
  • BIG PICTURE

Books

• M. L. Boas, Mathematical methods in Physical Sciences, 3rd edition, Wiley, 2006.
  

• G. B. Arfken, H. J. Weber, Mathematical methods for physicists, Academic press, London, 2001.
  

• G. Mahan, Applied Mathematics, Kluwer Academic/Plenum Publishers, New York, 2002.
  

• F. W. Byron and R. W. Fuller, Mathematics of classical and quantum physics, Dover Publications, Inc., New York, 1992.
  

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

Extra material

• Asymptotic methods, I (text) and II (text).
  

• Fourier and Laplace transforms, I (text) and II (text).
  

• Solving differential equations: overview + non-linear and one dimensions + Green's functions.
  

Homeworks

• Homework #1: due Friday, Sept. 29 (before the lecture). All problems are from Boas, 3rd edition. If asked to evaluate integral on computer, this means Mathematica.
  Chapter 11: Problems 3.15, 3.17, 7.8, 9.4, 10.2 (find 3 terms), 11.5, 12.3, 12.9, 12.21, 13.21.
Extra credit problem: Estimate the remainder of the series in 10.2 for p<4, x>0. Compare this estimate (absolute value) and the last term in expansion (3rd term) at large x.
Solutions


• Homework #2: due Friday, Oct. 6 (before the lecture).
  Problem #1: Kirchhoff's law for the resistance-inductance circuit is:  L*dI/dt + R*I = V, V = V₀ = constš is voltage, I=I(t) is current, R is resistance, and L is inductance. Find I(t)  if  I(t=0)=0.
Problems #2 through 10 -- Chapter 12: 1.2, 1.7, 2.3, 2.4, 3.4, 3.6, 4.3, 5.1, and 5.3.
Solutions


• Homework #3: due Friday, Oct. 13 (before the lecture).
Problem #1: Evaluate P_l'(x=1) (a) directly from the Rodrigues' formula; (b) from the generating function.
Problems #2 through 10 -- Chapter 12: 5.2, 5.12, 6.2, 6.4, 7.2, 8.3, 9.1, 10.8, 11.8.
Solutions


• Homework #4: due Friday, Oct. 20 (before the lecture).
Problems #1 through #5: Chapter 12: 12.9, 13.3, 15.3, 15.5, 16.8
Problem #6: Use Mathematica to plot  J_p(x)  for šp=0, 1, 2, 3. Plot them together with the corresponding asymptotic expressions for large and small x (see 12.20). For each regime identify the range of 10% accuracy of the corresponding asymptotic expression.
Problems #7 through #9: Chapter 12: 23.9, 23.10, 23.19.
Problem #10: Extra credit problem.
Solutions


• Homework #5    --->   Solutions
  
  
• Homework #6    --->   Solutions
  
  
• Homework #7: due Monday, Nov. 20 (before the lecture).
  Problems #1 through #10: Chapter 8: 11.8, 11.11, 11.15, 11.16, 11.21, 11.22, 12.1, 12.3, 12.6, 12.8
  Extra credit problem
  Solutions
  
  
• Homework #8    --->   Solutions
  
  

Exams

Academic honesty

Email

Mathematica, etc.