Qualifiers: Combined Analysis
This exam involves combining Complex Analysis with Real Analysis into one exam format.
Review of basic notions of complex numbers, Argand diagram, etc. Review of basic necessary topological properties of C.
Basic Complex Variables Theory:
Including Cauchy-Riemann equations, power series, integral formulae and their applications, (e.g., maximum modulus theorem, Schwarz's lemma, etc.)
Analytic continuation, residues, and applications, (e.g., Rouche's theorem, argument principle, etc.)
Including infinite products and applications, Mobius transformations, conformal mapping.
- MATH 62151 / 72151: Functions of a Complex Variable I
- MATH 62152 / 72152: Functions of a Complex Variable II
- L. Ahlfors, Complex Analysis, McGraw Hill.
- J. Conway, Functions of One Complex Variable, Springer-Verlag.
- W. Rudin, Real and Complex Analysis, McGraw Hill.
The student is expected to be familiar with those topics normally covered in a one-year, senior-level course in analysis. These topics include: real and complex numbers and their properties, properties of the metric space Rn, continuity, differentiability, Riemann integration, Riemann-Stieltjes integration, sequences and series of functions.
One-one functions, cardinal numbers, partial order, countability, Zorn's lemma.
Open and closed sets, convergent sequences, functions and continuity, semi-continuity, separable spaces, complete spaces, compact spaces, Baire category theorem.
Lebesgue outer measure, Borel sets, algebras, measurable sets and non-measurable sets, measure spaces.
Measurable Functions and Integration:
Properties, approximation by simple functions, Egoroff's theorem, monotone convergence theorem, convergence in measure, Fatou's lemma, Lebesgue dominated convergence theorem, signed measures, Lebesgue differentiation theorem, Hahn decomposition theorem, Radon-Nikodym theorem, Lebesgue decomposition theorem, Fubini's theorem, Tonelli's theorem, Stone-Weierstrass theorem, Ascoli-Arzela theorem.
Minkowski and Holder inequalities, Riesz representation theorem.
Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem.
Geometrical aspects, projection, Riesz-Fischer theorem.
- MATH 62051 / 72051:Functions of a Real Variable I
- MATH 62052 / 72052: Functions of a Real Variable II
- H. L. Royden, Real Analysis, MacMillan
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill