Detailed Outline

Mathematical Methods in Physics (PHY 230) - Fall 2003

• Introduction (1) [O1]
• Special Functions (1-2)
  • Elementary functions (1) [1.1]
  • Principal values (P or PP) (1) [H1]
  • Gaussian integrals (1)
  • Double factorials (1)
  • Error functions (erf, erfc) (1) [3.3, H1]
  • Gamma functions (1-2) [3.1, H1]
  • Beta functions (2) [3.2, H1]
  • Elliptic integrals and functions (2) [3.5, H1]
  • Riemann Zeta functions (2) [3.7, H1]
• Infinite Series (2-4)
  • Terminology and definitions (2) [1.2, 2.2]
  • Uniform convergence (2, L1) [2.5]
  • Convergence tests (reading) [2.3, 2.4, H2]
  • Standand series (zeta, geometric, exp, log, harmonic) (2) [2.2-6, H3]
  • Summing series (2-3)
  • Rational sums (3) [H4]
  • Power series (3) [2.6]
  • Expanding series (3) [2.7, H3]
  • Bernouilli numbers (3) [3.7, H3]
  • Asymptotic series (4, L1) [2.9, H5]
  • Convergence examples (reading) [H6]
• ODEs (4-9)
  • Definitions (4) [11, H7]
  • First order DEs - Separable (4) [11.1, H7]
  • First order DEs - Exact (4) [11.1, H7]
  • First order DEs - Integrating factors (4) [11.1, H7]
  • First order DEs - Linear eqn (5) [11.2, H7]
  • First order DEs - Bernoulli's eqn (5) [11.2, H7]
  • Second order DEs (5) [11.5, H7]
  • General DEs - Homogeneous/scale-invariant/isobaric eqn (5) [H7]
  • Linear DEs - Complementary and particular solutions (5) [11.3, H7]
  • Linear DEs - Constant-coefficient eqn (5) [11.3, H7]
  • Linear DEs - Euler-Cauchy eqn (5) [11.5, H7]
  • Linear DEs - Particular solutions - Methods (5)
  • Linear DEs - Undetermined coefficients (5) [11.4, H7]
  • Linear DEs - Variation of parameters (6) [11.4, H7]
  • Wronskian and linear independence (6) [11.1]
  • Linear DEs - Simultaneous ODEs (6) [11.6]
  • Series solutions - Introductions (6) [12.1]
  • Series solutions - Legendre's ODE (6) [12.1, 12.3, H8]
  • Series solutions - Legendre polynomials and functions (6-7) [12.3, H8]
  • Important ODEs (7) [H9]
  • Series solutions - Convergence (7) [12.2]
  • Series solutions - Classification (OP, RSP, ESP) (7) [12.2, H10]
  • Series solutions - Frobenius method (7) [12.4, H10]
  • Series solutions - Bessel function - J_m(x) (7-8, L2) [12.5-6, H11]
  • Series solutions - Second solution (8) [12.4, H7]
  • Series solutions - Bessel function - N_m(x) (8) [12.5-6, H12, O2]
  • Qualitative Methods - 1d flows (8, L3)
  • Qualitative Methods - 2d flows - phase plane (9, L3) [13.1, 13.4]
  • Qualitative Methods - Fixed points and classification (9) [13.2-3, O3, O4]
  • Qualitative Methods - 2d flows - closed orbits (11) [13.3]
  • Qualitative Methods - 3d flows - chaos (11) [13.5, O6]
  • Boundary conditions - Initial-value and boundary-value problems (9) [BoundaryValues.nb]
  • Boundary conditions - Eigenfunctions (9) [BoundaryValues.nb]
• Orthogonal Functions and Eigenfunctions (9-12)
  • Orthogonal sets of functions (9-10) [14.2, H13, H14]
  • Legendre Polynomials - Orthogonality & generating function (10, L4) [14.1, H8]
  • Orthogonal series (10) [14.1-2, Legendre.nb, O5]
  • Gram-Schmidt orthogonalization (10) [14.1]
  • Best approximations and Bessel's inequality (11-12) [14.1]
  • Completeness, Parseval's equality, and convergence in the mean (11)
  • Sturm-Liouville theorem (11) [14.3, H15]
  • Putting into Sturm-Liouville form (11) [14.3]
  • Bessel's ODE and Bessel Series (12) [12.6, 16.4, H11, Bessel.nb]
  • Linear vector spaces (12) [9.7]
  • Hermitean operators --> real eigenvectors and orthogonal eigenfunction (12) [10.2, 14.3]
  • Sturm-Liouville ODE and suitable BCs --> Hermitean (12) [14.3, H15]
  • Closure (14, L5) [H15]
• Delta functions (12-13)
  • Delta functions and delta sequences (12) [3.6, H16]
  • Transformed of variables (13) [H16]
  • 3-dimensional delta functions (13) [H16]
• Green's Functions (13-14)
  • Linear systems and Green's functions (13)
  • Green's functions - ODEs (13) [14.5, H17]
  • Matching method for Green's function (13-14, L5) [14.5, H17]
  • Using Green's functions with homogeneous BCs (example) (14) [14.5, H17]
  • Green's theorem (14)
  • Using Green's functions with general BCs (magic rule) (14) [H17]
  • Symmetry in Green's function for a Sturm-Liouville form (14) [H17]
  • Eigenfunction expansion of Green's functions (14, L5) [14.5, H17]
  • 3D Greeen's functions (14, L5) [H18]
• Orthogonal Coordinates (15)
  • Rectangular coordinates (15)
  • Cylindrical coordinates (15) [8.3, H19]
  • Orthogonal coordinates (15) [8.5, H21]
  • Spherical coordinates (15) [8.4, H20]
  • Vector components (15) [8.2, H19-H21]
  • grad, div, curl, Laplacian (15) [8.3-8.5, H19-H21]
• PDEs (16-18)
  • PDE -- Introduction, PDEs of physics, boundary conditions (16) [16.1, 16.7H22]
  • Techniques for solving PDEs (16)
  • Separation of variables (16) [16.2]
  • Laplace's equation - Rectangular coordinates (16-17, L6) [16.2, H23]
  • Laplace's equation - Cylindrical coordinates (17) [16.4, H9, H19, H23, O7]
  • Modified Bessel ODEs and functions (17) [12.5-6, H9, H24, H25, O7]
  • Laplace's equation - Spherical coordinates (17-18) [16.2, H20, H23]
  • Spherical Harmonics (18) [16.6, H23, SphericalHarm.nb, O9]
  • Wave equation, heat equation, Schrödinger equations (18) [16.6]
  • Helmholtz' equation (18) [16.1, H26]
  • Other separable systems (18) [8.6, O8]
• Integral Transforms (18-21)
  • Fourier series and orthogonal functions (18) [15.1-2, H27, H13]
  • From Fourier series to Fourier transforms (19) [17.5, H27, H28]
  • Integral transforms (19) [17.0]
  • Fourier transform - notations (19)
  • Fourier transform - Gaussian, uncertainty principle, shifting (19) [17.5, H28]
  • Fourier transform - (real & even) from/to (real & even) (19)
  • Fourier cosine transform (19)
  • Fourier transform - "proof" from delta function (19) [17.5]
  • Fourier transform - square pulse, sinc(x) (19) [17.5, H28]
  • Fourier transform - ODE's (20) [17.6, H28]
  • Convolution (20) [17.5, H28]
  • Power spectrum (20, L7)
  • Correlation and Wiener-Khinchin (20) [H28]
  • Fourier transform - Parseval's relation (20) [17.5, H28]
  • Laplace transform - Introduction (20) [17.1, H29]
  • Laplace transform - ODE's (21) [17.2-3, H29]
  • Laplace transform - PDE's (21) [17.4, H29]
• Functions of a Complex Variable (21-25)
  • Basic complex numbers and complex functions (21) [4, Complex.nb]
  • Multivalued functions, branch cuts, and branch points (21-22) [4.6, 18.1]
  • Limits, continuous functions and differentiable functions (22) [18.1-2, H30]
  • Cauchy-Riemaan relations (22) [18.2, H30]
  • Analytic functions (22) [18.2, H30]
  • Singularities (22) [18.2, H30]
  • Complex integration (22-23) [18.3]
  • Cauchy's theorem (23) [18.3, H30]
  • Cauchy's integral formula (23) [18.4, H30]
  • Power series and Taylor series (23) [18.5]
  • Laurent series (23-24) [18.5]
  • Residue theorem (24) [18.6, H30, H31]
  • Contour integration - ML lemma, and arc at infinity (24) [19.2, H31]
  • Contour integration - finding residues (24-25) [18.6, H31]
  • Contour integration - sinusoidal functions (25) [19.2, H31]
  • Contour integration - principal parts (25) [H31]
  • Contour integration - Jordan's lemma (25) [H31]
  • Contour integration - related paths (25) [H31]
  • Contour integration - log trick and branch cuts (25) [H31]
  • Contour integration - inverse Laplace transform (25) [19.1]
  • Conformal transformations (L8) [19.5-7]
• Probability (26-28)
  • Basic probability (26) [21.1, H32]
  • Permutations and combinations (26) [H33]
  • Binomial coefficients (26) [H33]
  • Random variables (26) [21.1]
  • Probability distributions (26) [21.1-2, H34]
  • Properties of distributions (26) [21.1-2, H34]
  • Joint distributions f(x,y) (27) [21.1-2, H34]
  • Continuous distributions - Gaussian (27) [21.2, H35]
  • Continuous distributions - Cauchy/Lorentzian (27) [H35]
  • Continuous distributions - Uniform (27) [H35]
  • Continuous distributions - Exponential and Gamma (27) [H35]
  • Transformed probability distribution (27)
  • Sums of random variables (27) [H35]
  • Binomial distribution (27) [21.1, H35]
  • Random events and Poisson processes (27) [21.5, O10]
  • Poisson distribution (27-28) [21.1, 21.5, H35]
  • Waiting time (28) [21.5, H35]
  • Gamma distribution (28) [21.5, H35]
  • Characteristic functions and moment generating functions (28) [21.3, H36]
  • Sums of random variables - convolution (28) [21.3, H36]
  • Central limit theorem (28) [21.3]