 Linear algebra: Recapitulation of basic concepts. Dual spaces, Eigenvalues, Eigenvectors, Similarity transformations, Diagonalization, Inner product spaces, Hilbert spaces.
 Complex analysis: Differentiability and analyticity, the CauchyRiemann conditions, the Cauchy theorem, Cauchy's residue theorem, Applications to the calculation of integrals and sums.
 Fourier and Laplace transforms: inversion of Fourier and Laplace transforms, applications to the solution of differential equations.
 SturmLiouville systems: Orthogonal polynomials, The hypergeometric and confluent hypergeometric functions, Bessel, Neumann and Hankel functions, Legendre and Hermite polynomials, Integral representations of special functions.
 Green's functions: For first and second order linear differential equations in one dimension, eigenfunction expansions, connection with the delta function, integral equations, perturbation theory based on Green's function.
 Group theory for the physical sciences: Review of the algebra of groups. Action of a group on a set. Action on function spaces. Matrix groups. Matrix representations. Reducible and irreducible representations. Schur's Lemma. Characters. Orthogonality relations. Regular representation. Character tables. Reduction of representations. Applications to crystallography, molecular vibrations and molecular orbital theory.
 Continuous and Lie groups: Irreducible representations of the groups SO(3) and SU(2). Infinitesimal group elements. Lie algebras. Simple and semisimple Lie groups and algebras. The Cartan subalgebra. Roots and weights. Classification of simple root spaces. Dynkin diagrams. Irreducible representations of su(3)  the quark model.
