This course introduces students to theories in Fourier analysis. Topics covered include: Fourier series, convergence of Fourier series, Cesaro and Abel summability, Plancherel formula, applications of Fourier series, Fourier transforms, inversion formula, Schwartz space, Poisson summation formula, applications of Fourier transform to partial differential equations, Fourier analysis on Z(N), Fourier analysis on finite abelian groups, Dirichlet’s theorem.
Topics covered include: analytic functions, power series, Goursat’s theorem, Cauchy integral formula, Morera’s theorem, Schwarz reflection principle, Runge’s approximation theorem, analytic continuation, zeros and poles, residue formula, meromorphic functions, argument principle, complex logarithm, harmonic functions, Jensen’s formula, functions of finite order, infinite products, Hadamard’s factorization theorem, Schwarz lemma, Riemann mapping theorem, gamma function, Riemann zeta function, elliptic functions, theta function.