Lectures:

11:30–12:20    Mondays, Wednesdays, and Fridays, Wean Hall Room 7316

Classes:

  3:30–4:20    Thursdays, Wean Hall Room 7316

Office hours:

by arrangement

Instructor:

Hael Collins, Wean Hall Room 7414

Grader:

Qi Fu, Wean Hall Room 7324

Textbook:

Sadri Hassani,
Mathematical Physics,
Springer-Verlag, New York, 1999.

Week I

August 29–Sept 2

Finite-dimensional vector spaces; the definition of a vector space; subsets and subspaces; examples; dimensions; linear transformations; matrices

Week II

September 7–9

The kernel of a map; injective maps; surjective maps; bijective maps; isomorphisms; similarity transformations; the inner product; definition of an inner product; the Gram-Schmidt orthogonalization procedure; the Schwarz Inequality

Week III

September 12–16

Commutators; Hermitian transformations; Hermitian conjugation; orthogonal transformations; unitary transformations; expectation values; projections; the completeness relation

Week IV

September 19–23

Permutations; placement of indices; tensors; types of matrices; the determinant; its definition and properties; how to invert a matrix; cofactors; the trace; the spectral decomposition

Week V

September 26–30

Eigenvalues and eigenvectors; characteristic polynomial; characteristic equation; eigenvalues of Hermitian matrices; normal matrices; the square-root of a matrix

Week VI

October 3–7

Infinite-dimensional vector spaces; Fourier analysis; Fourier transforms; the Dirac δ-function; properties of the δ-function; Fourier transforms in multiple dimensions; the Gibbs phenomenon

Week VII

October 10–14

Hilbert spaces; Cauchy sequences; complete vector spaces; Banach spaces; the Parseval Inequality; the Bessel Inequality; square-integrable functions; the Riesz-Fischer theorem; the Stone-Weierstrauss theorem

Week VIII

October 17–19

Recurrance relations; orthogonal polynomials; the Rodriguez formula; the classical orthogonal polynomials; Hermite polynomials; Laguerre polynomials; Jacobi polynomials; Chebyshev polynomials; generating functions

Week IX

October 24–28

Complex analysis; the complex plane; de Moivre’s theorem; complex functions; complex derivatives; the Cauchy-Riemann condition; complex analyticity; the complex exponential; conformal transformations

Week X

Oct 31–Nov 4

Examples of conformal transformations; complex integration; the Cauchy-Goursat theorem; curves and contours; the Cauchy Integral Formula; the Darboux inequality; entire functions

Week XI

November 7–11

Taylor and Laurent series; convergence; radius of convergence; tests of convergence; zeros and singularities; three types of singularities; residues; the Residue Theorem; complex integration

Week XII

November 14–18

Definite integrals of rational functions; definite integrals of products of rational and trigonometric functions; definite integrals of trigonometric functions; principal values

Week XIII

November 21

Meromorphic functions; the Mittag-Leffler expansion

Week XIV

Nov 28–Dec 2

Multivalued functions; Riemann surfaces; branch cuts; integrating around a branch cut; the method of steepest descent; the Sitrling approximation

Week XV

December 5–9

Linear differential equations; first-order linear equations; second-order linear equations; integrating factors; second-order linear operators on a Hilbert space; Green’s functions; Green’s identities; boundary-value data; completely homogeneous problems; Green’s functions in higher dimensions