Detailed Outline (RHB 1st edition)
Mathematical Methods in Physics (PHY 230)
- Fall
Numbers in parentheses
are the lectures
(n) or
the Computer Labs
(Ln).
Numbers in square brackets [...] are the
RHB1 sections (1st edition) and
handouts (Hn).
- Introduction (1)
- Special Functions (1)
- Gamma functions (1) [Appendix, H1]
- Beta functions (1) [Appendix, H1]
- Error functions (1) [Appendix, H1]
- Orthogonal Coordinates (1-2)
- Rectangular coordinates (1)
- Cylindrical coordinates (1-2) [8.9.1, H2]
- Orthogonal coordinates (2) [8.10, H4]
- Spherical coordinates (2) [8.9.2, H3]
- PDEs (2-5)
- PDE -- Example (2) [16.0, 16.1.2]
- PDEs of physics (3) [16.1, 16.3.3, 16.6.2, H5]
- Techniques for solving PDEs (3)
- Separation of variables (3) [17.1, 17.2]
- Helmholtz's equation - Rectangular coordinates (4) [17.2, H6]
- Helmholtz's equation - Cylindrical coordinates - Bessel functions (4) [17.3.0-4, H6, H7]
- Helmholtz's equation - Spherical coordinates - Spherical Harmonics
(4-5, L1) [17.3.0-4, H6, H7]
- Other separable systems (5)
- ODEs (5-9)
- Introduction (5) [12.0-1, 13.0-1.3]
- General DEs - Separable (5) [12.2.1, H8]
- General DEs - Exact (5) [12.2.2, H8]
- General DEs - Integrating factors (6) [12.2.3-4, H8]
- General DEs - Scale-invariant eqn (6) [12.2.5-7, 13.3.4-5, H8]
- General DEs - Special methods for 2nd-order DE (6) [13.3.1-2, H8]
- Linear DEs - Euler and Legendre Eqn (6-7) [13.2.1, H8]
- Linear DEs - Exact (7) [13.2.2, H8]
- Linear DEs - Reduction of order (7) [13.2.3, H8]
- Linear DEs - Particular integrals - Methods (7)
- Linear DEs - Variation of parameters (7) [13.2.4, H8]
- Wronskian and linear independence (7) [13.0]
- Series solutions - Introductions (7)
- Series solutions - Ordinary Points (7) [14.2]
- Series solutions - Legendre polynomial (7-8) [14.2, 14.5, 14.6, 14.6.1, H9]
- Classification (OP, RSP, ESP) and important ODEs (8) [14.1.1, H10]
- Fuchs' theorem and singular points (8) [14.1.1, 14.3, H10]
- Series solutions - Frobenius method (8) [14.3, H10]
- Series solutions - Bessel functions (8) [14.7-14.7.2]
- Methods for second solutions (9) [14.4]
- Wronskian method and Abel's formula (9) [14.1, 14.4.1]
- Convergence - Fuchs' theorem (9) [14.5, 14.6.0, 3.5.1, H10]
- Orthogonal Functions and Eigenfunctions (9-12)
- Orthogonal sets of functions (9) [15.1.0, 15.5, H12-13]
- Orthogonal series (9) [15.1.0, H12-13]
- Orthogonal series - Laplace equation in a box (9-10, L4) [17.2, H12-13]
- Orthogonal series - general boxes, cylinders, spheres (10) [17.3.2]
- Eigenfunctions and eigenvalues (10) [15.0]
- Sturm-Liouville theorem (10) [15.4, H14]
- Boundary conditions (10) [15.4.1 (wrong), H14]
- Putting into Sturm-Liouville form (10) [15.4.2]
- Completeness and approximation (11) [15.2]
- Gram-Schmidt orthogonalization (11) [15.1, 15.3.2]
- Linear spaces (11) [15.1]
- Hermitean operators --> real eigenvectors and orthogonal eigenfunctions [15.2-3, H14]
- Sturm-Liouville ODE and suitable BCs --> Hermitean (11) [15.4]
- Sturm-Liouville theorem - review (12) [H14]
- Legendre Polynomials - review and generating functions (12) [14.6.2, 15.5.1]
- Closure (14) [15.6]
- Green's Functions (12-14)
- Delta functions and delta sequences (12) [11.1.3]
- Linear problems and Green's functions (12)
- Construction from homogeneous solutions (ODEs) (matching method) (12) [13.2.5, H15]
- Using Green's functions (13) [13.2.5, H15]
- Hermitean operator - symmetry (13) [15.6, H12]
- Hermitean operator - general BCs (magic rule) (13) [17.5.2, H15]
- 3d Green's functions - Poisson equation (13) [17.5.0-2, H16]
- Poisson equation - fundamental solutions (14) [17.5.3]
- Poisson equation - method of images (14) [17.5.3]
- Eigenfunction expansion of Green's functions (14) [15.6-7, H15, H16]
- 3D Green's functions - 2 expansion + 1 matching method (14) [H17]
- Infinite Series (15-16)
- Terminology and definitions (14) [3.1, 3.3.0-1]
- Uniform convergence (termwise continuous/integration/differentiation)
(14, L5) [3.4]
- Convergence tests (15) [3.3.2, H18]
- Standand series (geometric, exp, harmonic, log, Riemann zeta) (15) [3.2.2, 3.3.2, H11]
- Summing series (15) [3.2, H19]
- Expanding series (16) [3.6, H11]
- Asymptotic series (L5, 16)
- Functions of a Complex Variable (16-21)
- Basic complex numbers and complex functions (16) [2, 18.4]
- Mapping (16, L6) [18.8]
- Multivalued functions, branch cuts, and branch points (16) [18.5]
- Continuous functions and differentiable functions (17) [18.1]
- Cauchy-Riemaan relations (17) [18.2]
- Analytic functions (17) [18.1, 18.4]
- Harmonic functions and complex potentials (17, L6) [18.2, 18.7]
- Conformal transformations (L6) [18.8-9]
- Complex integrals (17) [18.10, 9.1]
- Cauchy's theorem (17-18) [18.11]
- Cauchy's integral formula (18) [18.12]
- Isolated singularities (poles, essential) (18) [18.6]
- Branch points, and other singularities (18) [18.5-6]
- Power series and Taylor series (18) [18.3, 18.13]
- Analytic continuation and the identity theorem (19) [18.13]
- Laurent series (19) [18.13]
- Zeroes (19 [18.6]
- Residue theorem (19-20) [18.14]
- Contour integration - finding residues (20) [18.13, H20]
- Contour integration - arc at infinity (20) [18.17, H20]
- Contour integration - sinusoidal functions (21) [18.16, H20]
- Principal values (21) [18.17, H20]
- Contour integration - Jordan's lemma (21) [18.17, H20]
- Contour integration - related paths (21) [H20]
- Contour integration - log trick (21) [H20]
- Contour integration - branch cut (21) [18.18, H20]
- Integral Transforms (22-25)
- Fourier series and orthogonal functions (22) [10, H21]
- Parseval Theorem (22) [10.8, H21]
- Fourier transform - from Fourier series (22) [11.0, 11.1, H22]
- Fourier transform - notations (22)
- Integral transforms: (22)
- Fourier transform [11.1]
- Fourier sine/cosine transform [11.1.6]
- Laplace transform, and the Bromwich integral [11.2, 18.20]
- Handel transform [11.3]
- 3D Fourier transform [11.1.10]
- Fourier transform - "proof" from delta function (22) [11.1.4]
- Fourier transform - Gaussian, uncertainty principle (22) [11.1.1]
- Fourier transform - (real & even) from/to (real & even) (22) [11.1.6]
- Fourier transform - square pulse, sinc(x) (22) [11.1.2]
- Fourier transform - Parseval's relation (23) [11.1.9]
- Fourier transform - properties (23) [11.1.5, H22]
- Fourier transform - ODE's (23) [17.4, H22]
- Convolution and deconvolution (23) [11.1.7]
- Numerical methods and FFT (MMA-FFT)
- Transfer functions (23)
- Power spectrum (24, L7)
- Correlation and Wiener-Khinchin (24) [11.1.8, H22]
- Laplace transforms and the inverse (24) [11.2, 18.20, H23]
- Laplace transform - ODE's (24) [13.1.4, H23]
- Laplace transform - PDE's (24-25) [17.4, H23]
- Probability (25-27)
- Basic probability (25) [24.1-2]
- Permutations and combinations (25) [24.3, H24]
- Binomial coefficients (25) [24.3, H24]
- Probability distributions (25) [24.4, H25]
- Gaussian distribution (25-26) [24.8.1, H26]
- Uniform distribution (25-26) [26.8.6, H26]
- Properties of distributions (26) [24.5, H25]
- Transformed probability distribution y=h(x) (26) [24.10]
- Joint distributions f(x,y) (26) [24.11]
- Transformed probability distribution r=R(x,y) (26) [24.10]
- Sums of variates (26) [24.6.2]
- Binomial distributions (26) [24.7.1, H26]
- Random events and Poisson process (26) [24.7.3]
- Poisson distributions (26-27) [24.7.3, 24.8.2, H26]
- Waiting time (26-27) [24.8.5, H26]
- Gamma distributions (27) [H26]
- Binomial for (n to infinity) to Poisson or Gaussian (27) [24.8.2-3, H26]
- Random walks (27) [24.7.1, 24.8.2, H26]
- Generating functions (27) [24.6, H25-27]
- Sums of variates - Convolution (27) [24.6.2, H27]
- Central limit theorem (27) [24.9]
Handouts
Last updated: 26-Nov-02
