Detailed Outline

Mathematical Methods in Physics (PHY 230) - Fall 2004

Numbers in parentheses are the lectures (n) or the Computer Labs (Ln). Numbers in square brackets [] are the McQuarrie sections [n.m] or problems [n.m.p], or handouts [Hn], or overheads [On].

In a few places there are extra references beyond the book (McQuarrie) and the course. You don't need to read this, but maybe it's useful. The extra references are:

RHB1 = Riley, Hobson, and Bence, Edition 1
RHB2 = Riley, Hobson, and Bence, Edition 2

(See the Bibliography).

Introduction (1) [O1]
Special Functions (1-2)
- Elementary functions (1) [1.1]
- Principal values (P or PP) (1) [H1]
- Gaussian integrals (1)
  - Standard Gaussian integral (int e^-t^2 dt = sqrt(pi)) [3.1.10]
  - Trick: Dimensional analysis
  - Trick: Differentiate under the integral sign [3.4.11]
- Double factorials (1) [3.1.15, 3.1.16]
- Error functions (erf, erfc) (1) [3.3, H1]
- Gamma function (1-2) [3.1, H1, H2, O2]
  - [Euler] integral definition (1)
  - Recurrence relation (1)
  - Factorials (1)
  - Gamma(1/2) (1)
  - Extending the definition to negative x (2)
  - Complex z -- basic complex numbers (2) [Chap 4]
  - Stirling's Approximation and "Steepest Descent" (2)
- Beta functions (2) [3.2, H1]
- Exponential integrals (2) [3.4, H1]
- Elliptic integrals and functions (2) [3.5, H1]
- Riemann Zeta function (2) [3.7, H1]
  - Definition
  - Convergence - review [1.2, 2.2]
  - Euler's constant [3.4]
  - Example: Blackbody radiation/Phonons (tricks)
  - Geometric series [2.2]
Infinite Series (2-4)
- Standand series (geometric, exp, log, zeta, binomial) (2) [2.2-6, H3]
- Uniform convergence (L2) [2.5]
- Absolute convergence (3) [2.4]
- Power series (3) [2.6-8, H3]
  - Radius of convergence (absolute and uniform)
  - Standard series
  - Bernouilli numbers [3.7]
- Convergence tests (3) [2.3, 2.4, H4]
- Summing series (3-4)
  - Transform to known series (differentiatate, integrate, add/subtract, partial frctions, generalize) (3-4)
  - Difference method (3)
  - Rational sums (4) [H5]
- Asymptotic expansions and series (4) [2.9, O3, H5]
- Convergence examples (4) [H6]
ODEs (4-9)
- Definitions/Terminology (4) [11, H7]
- Constant-coefficient equations (4) [11.3, H7]
- First order ODEs (4-5) [11.1-2, H7]
  - Separable (4) [H7#1]
  - Exact (5) [H7#2]
  - Integrating factor (5) [H7#3]
  - Linear eqn (5) [H7#4]
  - Bernoulli's eqn (5) [H7#5]
  - x=F(dy/dx,y) (5) [H7#6]
  - y=F(dy/dx,x) (5) [H7#7]
- Second order ODEs (5) [11.5, H7]
  - No y [H7#8]
  - No x [H7#9]
- Homogeneous/scale-invariant/isobaric eqn (5) [H7#10-12]
- Miscellanous substitutions (5) [H7#14-16]
- Linear ODEs (6) [11.3-6, H7]
  - General solutions (complementary and particular) [H7#18]
  - Linear independence and Wronskains [9.5, 11.3]
  - Abel's formula [11.3.23]
  - Euler-Cauchy eqn [11.5, H7#21]
  - Particular solutions - Undetermined coefficients [11.4, H7#20]
  - Particular solutions - Variation of parameters [11.4, H7#24]
  - Systems of simultaneous ODEs [11.6]
- Series solutions (6-8) [12]
  - Legendre's ODE (6-7) [12.3, H8]
  - Legendre polynomials (7) [12.3, H8]
  - Legendre functions (7) [12.3]
  - Important ODEs (7) [14.2, H9]
  - Convergence - Fuchs' Theorem (7) [12.2, H10]
  - Classification (OP, RSP, ESP) (7) [12.2, H10]
  - Frobenius method - RSP (7-8) [12.4, H10]
  - Bessel's ODE (7-8) [12.5, H11]
  - Bessel function - J_m(x) (8) [12.5-6, H11]
  - Second solution (8) [12.4, H7#25]
  - Bessel function - N_m(x) (8-9) [12.5-6, H12, O4]
  - x₀ --> infinity (8)
- Qualitative Methods (9-10) [13]
  - 1d flows (9, L4) [Strogatz, 13.5]
  - Linear stability analysis - 1d (9) [Strogatz, 13.5]
  - 2d flows - fixed points and closed orbits (9, L4) [13.1, 13.4]
  - Linear stability analysis - 2d (9) [13.2-3]
  - Poincaré theorem - fixed points and classification (9) [13.3, O5]
  - Example - McQuarrie 13.3.8 (9) [13.3, O6]
  - 3d flows - fixed points, closed orbits, and strange attractors (10)
  - Lorenz Equations and chaos (10) [13.5, O7]
- Boundary conditions (10) [14.3, BoundaryValues-F04.nb]
  - Initial-value and boundary-value problems
  - Homogeneous boundary conditions
  - Eigenvalues and eigenfunctions
Orthogonal Functions and Eigenfunctions (10-12) [14]
- Orthogonal sets of functions (10) [14.2, H13, H14]
- Orthogonal series (10-11) [14.1-2, 14.4]
  - Fourier Sine Series (10) [H14]
  - Legendre Series (10) [H8, O8, Legendre-F04.nb]
  - Convergence -- Best approximations, convergence in the mean, completeness, Bessel's inequality, Parseval's equality (11)
- Gram-Schmidt orthogonalization (11) [14.2]
- Legendre Polynomials (11-12, L5) [Legendre-F04.nb]
  - Explicit polynomials (11) [14.1]
  - Rodrigues' formula (11) [H8]
  - Recurrence relations (11) [14.1, H8]
  - Generating function (12, L5) [14.1, H8]
- Complex vector spaces and Hermitean operators (12) [9.7]
- Hermitean operators (12)
  - Complex vector spaces and inner products [9.7]
  - Adjoint operators and Hermitean operators [14.3]
  - Hermitean operators --> real eigenvectors and orthogonal eigenvectors [10.2, 14.3]
- Sturm-Liouville theory (12-13) [14.3, H15]
  - Sturm-Liouville problem
  - Sturm-Liouville problem --> Hermitean operators
  - Suitable boundary conditions, and regular, periodic, singular
  - Integrating factor --> Sturm-Liouville form (13)
  - Bessel's ODE and Bessel Series (13, Bessel-F04.nb) [12.6, 16.4, H12]
  - Closure (14, L6)
Delta functions (13)
- Delta functions and delta sequences (13) [3.6, H16]
- Transformed of variables (13) [H16]
- 3-dimensional delta functions (13) [H16]
Green's Functions (13-15)
- Linear systems and Green's functions (13)
- Green's functions for ODEs (13-14) [14.5, H17]
  - Introduction (13)
  - Matching method for Green's function (14, L5) [14.5, H17, RHB1 13.2.5/RHB2 15.2.5]
  - Using Green's functions with homogeneous BCs (example) (14) [14.5, H17]
  - General boundary conditions and Green's theorem (14) [H17]
  - Eigenfunction expansion of Green's functions (14, L6) [14.5, H17, RHB1 15.6-7/RHB2 17.6-7]
- 3D Green's functions - Poisson equation (14-15, L6) [H18, RHB1 17.5.0-3/RHB2 19.5.0-3]
  - Introduction (14)
  - Boundary conditions -- Dirichlet and Neumann BCs (15)
  - Fundamental solution + Laplace equation (15)
  - Method of Images (15)
  - Matching method and/or Eigenfunctions (15) [O9, O10]
Orthogonal Coordinates (15-16)
- Rectangular coordinates (15)
- Cylindrical coordinates (15-16) [8.3, H19]
  - Definition, inverse, and intervals
  - Coordinate curves and coordinate surface
  - Base vectors, unit vectors, and scale factors
  - Infinitesimal displacment, arc length, volume, Jacobian
- General coordinates, orthogonal coordinates, and the metric tensor (16) [8.5, H21]
- Spherical coordinates (16) [8.4, H20]
- Vector components (16) [8.2, H19-H21]
- Vector operators (grad, del, curl, Laplacian) (16) [8.3-8.5, H19-H21]
PDEs (16-19)
- PDEs of physics (16-17) [16.1]
- Classification and boundary conditions (17) [16.7, H22]
- Techniques for solving PDEs (17)
- Separation of variables (17-19) [16.2]
  - Introduction (17)
  - Laplace's equation - Cartesian coordinates (17) [16.2, H23]
    - Cubical box (17-18, L7) [16.2]
  - Laplace's equation - Cylindrical coordinates (18) [16.4, H9, H19, H23, O11]
    - Modified Bessel ODEs and functions (18) [12.5-6, 16.2, H9, H24, H25, O11]
    - Cylindrical box (18) [16.2]
  - Laplace's equation - Spherical coordinates (18-19) [16.2, H20, H23]
    - Associated Legendre Functions (18-19) [16.6, SphericalHarm-F04.nb, O12]
    - Spherical Harmonics (19) [16.6, SphericalHarm-F04.nb, O12]
    - Angular momentum operators (19)
  - Wave and heat equations (19) [16.3-5]
  - Helmholtz' equation (19) [16.1, H26]
    - Spherical Bessel functions (19) [see p. 616 and section 16.8!]
  - Other separable systems (19) [8.6, O13]
Integral Transforms (19-22)
- Fourier series and orthogonal functions (19) [15.1-2, H27, H13]
- Fourier transforms (20-22) [17.5-6, H28]
  - From Fourier series to Fourier transforms (20) [17.5, H27, H28]
  - Convergence (20) [17.5]
  - Notations (20)
  - Gaussian, uncertainty principle, shifting (20) [17.5, H28]
  - (Real & even) from/to (real & even) (20)
  - Fourier cosine transform (20)
  - Fourier integral theorem (20) [17.5]
  - Square pulse, sinc(x) (20) [17.5, H16, H28]
  - ODEs (20-21) [H28]
  - PDEs (21) [17.6, H28]
  - Convolution (21)
    - Definition [17.5, H28]
    - Instrumental resolution
    - Convolution theorem [17.5, H28]
    - Transfer function
    - Windowing
  - Power spectrum density (PSD) (21)
    - Definition
    - Discrete and continuous spectra
    - log-log -- often "scale free"
    - Parseval's relation [17.5, H28]
  - Correlation and Wiener-Khinchin (21) [H28]
    - Auto- and cross-correlation definitions
    - Correlation theorem
    - Wiener-Khinchin theorem
  - 2D and 3D Fourier Transforms (22) [17.5]
- General integral transforms (22) [17.0, H29]
- Laplace transform (22)
  - Definitions [17.1, 17.7, 19.1, H30]
  - Fourier and Laplace Transforms
  - ODEs [17.2-3, H30]
  - PDEs [17.4, H30]
Functions of a Complex Variable (23-26)
- Complex numbers and complex functions (Read) [4, H31, Complex-F04.nb]
- Multivalued functions, branch cuts, and branch points (23) [4.6, 18.1, H31]
- Limits, continuity and differentiation (23) [18.1-2, H31]
- Cauchy-Riemann relations (23) [18.2, H31]
- Analytic functions (23) [18.2, H31]
- Singularities (23) [18.2, H31]
  - Poles
  - Isolated essential singularities
  - Non-isolated singularities---branch points, accumulate poles, other
  - Removable singularities
  - Point at infinity
- Power series and Laurent series (24) [18.5, H31]
- Complex integration (24) [18.3]
- Cauchy's theorem (24) [18.3, H31]
- Residue theorem (24) [18.6, H31, H32]
- Contour integration (24-26)
  - ML lemma, and arc at infinity (25) [19.2, H32]
  - Finding residues (25) [18.6, H32]
  - Jordan's lemma (25) [H32]
  - Sinusoidal functions (25) [19.2, H32]
  - Related paths (26) [H32]
  - Log trick and branch cuts (26) [H32]
  - Principal parts (26) [H32]
  - Inverse Laplace transform (26) [19.1]
- Cauchy's integral formula (26) [18.4, H31]
- Laurent series (26) [18.5, H31]
- Conformal mapping (L8) [19.5-7]
Probability (27-29)
- Basic probability (27) [21.1, H33]
- Permutations and combinations (27) [H34]
- Binomial coefficients (27) [H35]
- Probability distributions (27-28) [21.1-2]
  - Random variables (27) [21.1]
  - Definitions (pmf and pdf) (27) [H36]
  - Mean, E[X], moments, variance (27) [21.1-2, H36]
  - Transformed of probability distribution (27-28)
- Joint distributions (28) [21.1-2, H36]
- Important probability distributions (28) [H37]
- Binomial distribution (28) [21.1, H37]
- Random events
  - Poisson events (28) [21.5]
  - Poisson distribution (28) [21.1, 21.5, H37]
  - Waiting time (28-29) [21.5]
    - Exponential distribution (28) [H37]
    - Gamma distribution (29) [H37]
- Characteristic functions and moment generating functions (29) [21.3, H38]
- Sums of random variables - convolution (29) [21.3, H38]
- Central limit theorem (29) [21.3]