| week | subject |
|---|---|
| [01] Feb.03 |
Differential equations of hypergeometric type, Polynomial functions of hypergeometric type, The Rodrigues formula |
| [02] Feb.07 Feb.10 |
Eigenfunctions and eigenvalues of the Legendre and Laguerre differential equations, Solution of the one dimensional harmonic oscillator wave equation: Hermite polynomials, Raising, lowering and number operators for wavefunctions of a harmonic oscillator potential, Notes on functions of the complex variable z, The Cauchy-integral formula for a regular function f(z) and for its derivatives, Schlafli's and Laplace's integral representations of Legendre polynomials, The generating function of Legendre polynomials, The Legendre polynomial expansion of the electrostatic potential of a point charge and an electric dipole, Solution of Laplace's equation in spherical geometries with azimuthal symmetry -- The method of separation of variables. |
| [03] Feb.14 Feb.17 |
The residue theorem, Contour integration, Generating function of polynomial solutions of a differential equation of hypergeometric type, Generating functions of Legendre, Laguerre and Hermite polynomials, Recursion relations between the Legendre polynomials and their derivatives derived from the generating function, |
| [04] Feb.21 Feb.25 |
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| [05] Feb.28 Mar.03 |
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| [06] Mar.07 Mar.10 |
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| [07] Mar.14 Mar.17 |
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| [08] Mar.21 Mar.24 |
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| [09] Apr.04 Apr.07 |
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| [10] Apr.11 Apr.14 |
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| [11] Apr.18 Apr.21 |
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| [12] Apr.25 Apr.28 |
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| [13] May.02 May.05 |
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| [14] May.09 May.12 |
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| [15] May.18 |
Wave motion in inhomogenous media, Lateral vibrations of a rotating chain and its eigenfrequencies, The residue theorem, The generating function of polynomials of hypergeometric type derived from the Cauchy integral formula, The generating functions of the Laguerre and Hermite polynomials. Orthogonality of polynomials of hypergeometric type, Recursion relations of orthogonal polynomials, Recurrence relations of Legendre polynomials derived from the Rodriguez' formula and the generating function, Norm of Legendre polynomials, Gauss numerical quadrature for integration -- Relevance to the classical orthogonal (Legendre, Laguerre, Hermite) polynomials. Vibrational eigenmodes of a hanging chain, Series solution of the Bessel differential equation, Generating function, recursion relations and integral representation of Bessel functions of the first kind, The iterative method of Stodola and Vianello for the approximate determination of eigenvalues and eigenfunctions. Sturm-Liouville boundary value problems, Orthogonality, Expansion of arbitrary functions in series of orthogonal functions, Bessel's inequality, Completeness and Parseval's identity, Two dimensional classical wave equation, Vibrational eigenmodes of a circular drum membrane of uniform mass density and surface tension. Convergence of the method of Stodola and Vianello, Variational approximation and eigenvalue problems, Variational estimate of the lowest mode frequency for a circular drum head, Variational solution of the ground state energy of a particle in a triangular potential well: Fong-Howard trial wavefunction, Airy functions Variational calculation of the electrostatic potential due to a given fixed charge density, The Gamma function: definite-integral, infinite-product and infinite-limit representations, Stirling's asymptotic formula, Digamma and polygamma functions, The Beta function. Evaluation of some definite integrals in terms of the Beta function Incomplete gamma functions, The error function, The exponential, sine, cosine and logarithmic integrals, Elliptic integrals of the first and second kinds, Some elementary techniques on evaluation of integrals, Notes on evaluation of series, The hypergeometric function. The confluent hypergeometric (Kummer) function, Kummer transformation, Representations of some elementary functions and special polynomials in terms of the Kummer function; e.g., the incomplete gamma and error functions, Laguerre polynomials, Bessel and modified Bessel functions. Conventional nondegenerate time-independent perturbation theory, Eigenfrequency correction for a nonuniform string, Eigenfrequencies of vibration of a circular drum head with nonuniform mass density; e.g. the case of a point mass fixed on the membrane and a drum head of concentric membranes of different mass densities. The Dirac delta function, representations in terms of curvilinear coordinates, The Laplacian of the inverse-r potential, The Helmholtz wave equation, Spherical Bessel functions and their orthogonality relations, Solution of Laplace's equation in cylindrical coordinates, Modified Bessel functions Stoke's and divergence theorems, Green's identities, Green's function, Formal solution of Dirichlet and Neumann type boundary value problems using the Green's function Integral equations - relation to differential equations, Types: Fredholm and Volterra equations, Solutions of Fredholm integral equations for separable kernels, Orthogonality of eigenfunctions, Infinite series solutions |