Detailed Outline

Mathematical Methods in Physics (PHY 230) - Fall 2005

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:

• Strogatz, Nonlinear Dynamics and Chaos
• RHB1 = Riley, Hobson, and Bence, Edition 1, Mathematical Methods...
• RHB2 = Riley, Hobson, and Bence, Edition 2, Mathematical Methods...

• Introduction (1) [O1]
• Special Functions (1-3)
  • Elementary functions (1) [1.1]
  • Principal values (P or PP) (1) [H1]
  • Error functions (erf, erfc) (1) [3.3, 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]
  • Gamma function (1-2) [3.1, H1, H2, O2]
    • [Euler] integral definition (1)
    • Recurrence relation (1)
    • Factorials (1)
    • Gamma(n/2) (2)
    • Extending the definition to negative x (2)
    • Reflection formula, duplication formula (2)
    • Digamma & polygamma functions and Euler's constant (2)
    • Complex z and 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 (2) [3.5, H1]
  • Delta functions (2-3) [H3]
    • Discrete and continuous delta "functions" (2) [3.6]
    • Delta sequences (2) [3.6]
    • Integral represention (3)
    • Transformed of variables (3)
    • 3-dimensional delta functions (3)
• Infinite Series (3-4)
  • Definitions and examples: (3)
    • Convergent and divergent series [1.2, 2.2]
    • Harmonic series (divergent), alternating harmonic series (convergent) [2.4]
    • Uniform convergence (L2) [2.5]
    • Absolute convergence [2.4]
  • Standand series: (3)
    • Geometric series [2.2, H4]
    • Binomial series [2.7, H4]
    • Exponential series [H4]
    • Log series [2.6, H4]
    • Taylor/Maclaurin series/expansion [2.7-8]
    • Riemann zeta function/series [2.3, 3.7, H4]
  • Other power series: (3)
    • By integration [2.6, 2.8, H4]
    • Bernoulli numbers [3.7, H4]
  • Convergence tests (4) [2.3, 2.4, H5]
    • Integral test
    • Ratio test
    • Liebnitz' alternating series test
  • Summing series (4)
    • Transform to known series (differentiatate, integrate, add/subtract, partial fractions, generalize)
    • Difference method
    • Rational sums [H6]
    • Trigonometrical series
  • Asymptotic expansions (4-5) [2.9, O3]
    • Example: erfc(x)
    • Asymptotic series
• ODEs (5-9)
  • Definitions/Terminology (5) [11, H7]
  • General ODEs
    • First order ODEs (5) [11.1-2, H7]
      • Separable [H7#1]
      • Exact [H7#2]
      • Integrating factor [H7#3]
      • Linear eqn [H7#4]
      • Bernoulli's eqn [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]
      • Initial-value and boundary-value problems
      • No y [H7#8]
      • No x [H7#9]
    • Homogeneous/scale-invariant/isobaric eqn (6) [H7#10-12]
    • Miscellanous substitutions (6) [H7#14-16]
  • Linear ODEs (6-8) [11.3-6, H7]
    • General solutions (complementary and particular) (6) [H7#18]
    • Linear independence and Wronskains (6) [9.5, 11.3]
    • Abel's formula (6) [11.3.23, H7#26]
    • Complementary solutions (6)
      • Constant-coefficient equations [11.3, H7#19]
      • Euler-Cauchy eqn [11.5, H7#21]
      • Legendre eqn [H7#22]
      • Reduction of Order [11.3, H7#24]
    • Particular solutions (6)
      • Undetermined coefficients [11.4, H7#20]
      • Variation of parameters [11.4, H7#25]
    • Systems of simultaneous ODEs (7) [11.6]
    • Important ODEs and functions (7) [12.3-6, 14.1-2, 16.6, H8]
    • Series solutions (7-9) [12]
      • Legendre's ODE (7) [12.3, H9]
      • Legendre polynomials (7) [12.3, H9]
      • Legendre functions (8) [12.3]
      • Convergence for power series (8) [2.6, 18.5]
      • Fuchs' Theorems (8) [12.2-4]
      • Classification (OP, RSP, ISP) (8) [12.2, H10]
      • Frobenius method - RSP (8) [12.4, H10]
      • Bessel's ODE (8) [12.5, H11]
      • Bessel function - J_m(x) (8-9) [12.5-6, H11]
      • Second solution (9) [12.4, H7#26]
      • Bessel function - N_m(x) (9) [12.5-6, H11, O4]
      • x₀ --> infinity (9)
  • Qualitative Methods (9-10) [13]
    • Indroduction (9)
    • 1d flows (9, L4) [Strogatz, 13.5, O5]
    • 2d flows - fixed points and closed orbits (9, L4) [13.1, 13.4, O5]
    • Linear stability analysis (9-10) [13.2-3]
    • Poincaré theorem - fixed points and classification (10) [13.3, O6]
    • 3d flows - fixed points, closed orbits, and strange attractors (10)
    • Lorenz Equations and chaos (10) [13.5, O7]
• Orthogonal Functions and Eigenfunctions (10-13) [14]
  • Eigenvalues and eigenfunctions (10) [14.4]
  • Orthogonal sets of functions (10) [14.2, H12, H13]
  • Orthogonal series and generalized Fourier coefficients (10) [14.1-2, 14.4]
  • Example: Legendre Series (10) [H9, O8, Legendre-F05.nb]
  • Convergence: (11) [14.1-2, 14.4]
    • Best approximations
    • Bessel's inequality
    • Parseval's equality
    • Convergence in the mean
    • Completeness
  • Closure (11)
  • Sturm-Liouville theory (11) [14.3, H14]
  • Integrating factor --> Sturm-Liouville form (11) [14.3]
  • Complex vector spaces and 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 problem --> Hermitean operators
  • Suitable boundary conditions, and regular, periodic, singular
  • Gram-Schmidt orthogonalization (13) [14.2]
  • Legendre Polynomials (13, L5, Legendre-F05.nb/MMA)
    • Explicit polynomials (MMA) [14.1]
    • Rodrigues' formula (MMA) [H9]
    • Recurrence relations (MMA) [14.1, H9]
    • Generating function (13) [14.1, H9]
    • Legendre and Chebyshev series (10, O8, L5, MMA)
• Green's Functions (13-15)
  • Introduction (13-14)
  • Matching method (14) [14.5, H16, RHB1 13.2.5/RHB2 15.2.5]
  • Using Green's functions (example) (14) [14.5, H16]
  • Eigenfunction method (15, L6) [14.5, H16, RHB1 15.6-7/RHB2 17.6-7]
• Orthogonal Coordinates (15-16)
  • Cylindrical, spherical, and general coordinates (15) [8.3-5, H17-19]
  • Definitions, inverses, and domains (15) [8.3-5, H17-18]
  • Coordinate curves and coordinate surfaces (15) [8.6, H17-18]
  • Base vectors, unit vectors, and scale factors (15) [8.3-6, H17-19]
  • Infinitesimal displacment, arc length, volume, Jacobian (15) [8.3-6, H17-19]
  • Metric tensor (16)
  • Vector components (16) [8.2, H17-19]
  • Vector operators (grad, del, curl, Laplacian) (16) [8.3-8.5, H17-19]
• PDEs (16-19)
  • PDEs of physics (16)
    • Classification and boundary conditions [16.7, H20]
    • Laplace and Poisson equations [16.1-2]
    • Heat/diffusion equation [16.1, 16.5]
    • Wave equation [16.1, 16.3-4]
    • Schrödinger equations [16.1, 16.6]
    • Other PDEs
  • Techniques for solving PDEs (16)
  • Separation of variables (16-19)
    • Laplace's equation - Cartesian coordinates (16-17) [16.2, H21]
      • Cubical box (17, L7) [16.2]
    • Laplace's equation - Cylindrical coordinates (17) [16.4, H11, H17, H21, O10]
      • Modified Bessel ODEs and functions - I_m(x) and K_m(x) (17) [12.5-6, 16.2, H8, H21, H22, O10]
      • Cylindrical box (18) [16.2]
    • Separable systems (18) [8.3, O11]
    • Laplace's equation - Spherical coordinates (18-19) [16.2, H18, H21]
      • Associated Legendre Functions (18) [16.6, SphericalHarm-F05.nb, O10]
      • Spherical Harmonics (18) [16.6, SphericalHarm-F05.nb, O10]
      • Laplace series (18)
    • Wave, heat/diffusion, and Schrödinger equations (19) [16.3-6]
    • Helmholtz' equation (19) [16.1, H23]
      • Spherical Bessel functions (19) [see p. 616 and section 16.8!]
• Integral Transforms (19-22)
  • Fourier series and orthogonal functions (19) [15, H24, H12]
  • Fourier transforms (19-22)
    • From Fourier series to Fourier transforms (19) [17.5]
    • Fourier integral theorem (20) [17.5]
    • Convergence (20) [17.5]
    • Notations (20)
    • Gaussian and uncertainty principle (20) [17.5, H25]
    • Square pulse, sinc(x) (20) [17.5, H3, H25]
    • f(t)=1 (20)
    • (Real & even) from/to (real & even) (21)
    • Fourier cosine transform (21)
    • General integral transforms (21) [17.0, H26]
    • Tables - time shifting (21) [H25]
    • ODEs and PDEs (21) [17.6, H25]
    • Parseval's relation [17.5, H25]
    • Convolution (21)
      • Definition [17.5, H25]
      • Instrumental resolution [FFT-F05.nb]
      • Convolution theorem [17.5, H25]
      • Transfer function [FFT-F05.nb]
    • 2D and 3D Fourier Transforms (21) [17.5]
    • Power spectrum density (PSD) (21-22)
      • Definition
      • Discrete and continuous spectra
      • log-log -- often "scale free" -- Brownian and White noise
      • Total power
    • Correlation and Wiener-Khinchin (22) [H25]
      • Auto- and cross-correlation definitions
      • Correlation theorem
      • Wiener-Khinchin theorem
  • Laplace transforms (22)
    • Definitions [17.1, 17.7, 19.1, H26, H27]
    • Fourier and Laplace Transforms
    • ODEs [17.2-3, H27]
    • PDEs [17.4, H27]
• Functions of a Complex Variable (22-26)
  • Complex numbers and complex functions (Read) [4, H28, Complex-F05.nb]
  • Multivalued functions, branch cuts, and branch points (22-23) [4.6, 18.1, H28]
  • Limits and continuity (23) [18.1, H28]
  • Differentiation (23) [18.2, H28]
    • Cauchy-Riemann relations
    • Analytic functions
  • Power series (23) [18.5, H28]
  • Singularities (23) [18.2, H28]
    • Poles
    • Isolated essential singularities
    • Non-isolated singularities---branch points, accumulate poles, other
    • Removable singularities
  • Complex integration (23-24) [18.3]
  • Cauchy's theorem (24) [18.3, H28]
  • Integrals around poles (24) [18.6]
  • Residue theorem (24) [18.6, H28, H29]
  • Definite integrals (24-25)
    • ML lemma, and arc at infinity (24) [19.2, H29]
    • Sinusoidal functions (24) [19.2, H29]
    • Finding residues (24) [18.6, H29]
    • Jordan's lemma (24) [H29]
    • Related paths (25) [H29]
    • Principal values (25) [H29]
    • Log trick and branch cuts (25) [H29]
    • Inverse Laplace transform (25) [19.1]
  • Cauchy's integral formula (26) [18.4, H28]
  • Laurent series (26) [18.5, H28]
  • Conformal mapping (26, L9) [19.5-7]
• Probability (26-29)
  • Basic probability (26) [21.1, H30]
  • Permutations and combinations (26-27) [H31]
  • Binomial coefficients (27) [H32]
  • Random variables (27) [21.1]
  • Probability distributions (27-28) [21.1-2]
    • Definitions (pmf and pdf) and important probability distributions (27) [H33, H34]
    • Mean, <x>, E[X], <f(x)>, moments, variance, sd, chacteristic fn, MGF (27) [21.1-2, H33]
    • Transforms of probability distributions (two ways) (27)
    • Joint distributions (28) [21.1-2, H33]
  • Binomial distribution (28) [21.1, H34]
  • Random events (28)
    • Poisson events [21.5]
    • Poisson distribution [21.1, 21.5, H34]
    • Waiting times [21.5]
      • Exponential distribution [H34]
      • Gamma distribution [H34]
  • Characteristic functions (28-29) [21.3, H35]
  • Sums of random variables - convolution (29) [21.3, H35]
  • Central limit theorem (29) [21.3]