Fourier Series: definition, Fourier exponential series, trigonometric Fourier series; Fourier Transform: Integral transform, Fourier integral, inverse Fourier transform, properties of Fourier transform, Perseval’s theorem; Poisson’s summation formula, Fourier transform in Rd; Finite Group: group axiom, the finite group at low order, permutation group, simple groups, Young tableaux; Finite Group Representations: introduction, Schur’s lemma, Kronecker products, real and complex representation; Group Character: definition, projection formula, examples, Frobenius’s character formula; SU(2): introduction, some representations, from Lie algebra to Lie groups, applications; Beta and Gamma Function: beta and gamma integrals, Euler product, reflection formula, duplication formula, asymptotics, digamma function, Hurwitz and Riemann zeta function; Series Solution of Second Order ODEs: Frobenius’s method, series around a regular point, expansion around a regular singular point; Bessel Function: differential equation, generating function, Hankel function, asymptotic expansion; Legendre Polynomial: differential equation, Legendre function of the first and second kind, generating function, associated Legendre polynomial, spherical harmonics; Hypergeometric Function: hypergeometric series, Euler’s integral representation, the hypergeometric equation, contiguous relations, quadratic transformations, transformation and some special values; Sturm-Liouville Theory: orthogonal polynomials, examples, Hermitian operators, generalized Fourier series.