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Analysis

Course material: Mathematics 245AB, the first half of Mathematics 245C, and Mathematics 246AB.

Real Analysis Topics: Lebesgue integration; convergence theorems (uniform convergence, Ego- roff’s theorem, Lusin’s theorem, Lebesgue dominated convergence theorem, monotone conver- gence theorem, Fatou’s lemma); Fubini’s theorem; covering lemmas and differentiation of measures (Lebesgue decomposition theorem, Radon-Nikodym theorem, Jordan decomposition theorem, rela- tions to functions of bounded variation, signed measures and Hahn decompositions); approximate identities; basic functional analysis (linear functionals, Hahn-Banach theorem, open mapping the- orem, closed graph theorem, uniform boundedness principle, strong, weak, and weak* topologies); elementary point set topology including Urysohn’s lemma, the Tychonoff theorem, the Baire Cat- egory theorem and the Stone-Weierstrass theorem. The spaces C(X), the Riesz representation theorem, and the compact subsets of C(X), (Arzela-Ascoli theorem); Hilbert space, self-adjoint linear operators and their spectra; Lp spaces (duality, distribution functions, weak Lp spaces, Hölder’s inequality, Jensen’s inequality, linear operators); basic Fourier analysis (orthonormal sys- tems, trigonometric series, convolutions on Rn, Plancherel’s theorem, Riemann-Lebesgue lemma Poisson summation formula); abstract measure theory; Hausdorff measures.

Complex Analysis Topics: Analytic functions: Examples, sums of power series, exponential and logarithm functions, M¨obius transformations, and spherical representation. Cauchy’s theorem: Goursat’s proof, consequences of Cauchy integral formula, such as Liouville’s theorem, isolated singularities, Casorati-Weierstrass theorem, open mapping theorem, maximum principle, Morera’s theorem, and Schwarz reflection principle. Cauchy’s theorem on multiply connected domains, residue theorem, the argument principle, Rouch´e’s theorem, and the evaluation of definite inte- grals. Harmonic functions: conjugate functions, maximum principle, mean value property, Poisson integrals, Dirichlet problem for a disk, Harnack’s principle, Schwarz lemma and the hyperbolic metric. Compact families of analytic and harmonic functions: series and product developments, Hurwitz theorem, Mittag-Leffler theorem, infinite products, Weierstrass product theorem, Poisson- Jensen formula. Conformal mappings: Elementary mappings, Riemann mapping theorem, mapping of polygons, reflections across analytic boundaries, and mappings of finitely connected domains. Subharmonic functions and the Dirichlet problem. The monodromy theorem and Picard’s theorem. Elementary facts about elliptic functions.

Note: Also all material for the Basic Examination. To prepare, students are advised to work problems from as many old examinations as possible.

References

  • Folland, G.B. (1984). Real Analysis, New York, Wiley.
  • Roydan, H.L. (1969). Real Analysis, New York, MacMillan
  • Rudin, W. (1986). Real and Complex Analysis, New York, McGraw Hill (3rd edition).
  • Stein, E.M. and Sharkarchi, R. (2005). Real Analysis, Measure Theory, Integration and Hilbert Spaces, Princeton University Press.
  • Wheeden, R. and Zygmund, A. (1977). Measure and Integral, An Introduction to Real Analysis, New York, M. Dekker.
  • Ahlfors, L. (1979). Complex Analysis, New York, McGraw Hill (3rd edition).
  • Gamelin, T.W. (2001). Complex Analysis, New York, Springer.
  • Stein, E.M. and Sharkarchi, R. (2003). Complex Analysis, Princeton University Press
  • Rudin, W. (1976). Principles of Mathematical Analysis, New York. McGraw Hill. (3rd edition).
  • Stein, E.M. and Sharkarchi, R. (2003). Fourier Analysis, Princeton University Press.