Qualifiers: Combined Analysis
This exam involves combining Complex Analysis with Real Analysis into one exam format.
Analysis
Preliminaries:
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.)
Singularities:
Analytic continuation, residues, and applications, (e.g., Rouche's theorem, argument principle, etc.)
Additional Topics:
Including infinite products and applications, Mobius transformations, conformal mapping.
Suggested Courses:
- MATH 62151 / 72151: Functions of a Complex Variable I
- MATH 62152 / 72152: Functions of a Complex Variable II
Suggested References:
- L. Ahlfors, Complex Analysis, McGraw Hill.
- J. Conway, Functions of One Complex Variable, Springer-Verlag.
- W. Rudin, Real and Complex Analysis, McGraw Hill.
Real Analysis
Preliminaries:
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.
Set Theory:
One-one functions, cardinal numbers, partial order, countability, Zorn's lemma.
Metric Spaces:
Open and closed sets, convergent sequences, functions and continuity, semi-continuity, separable spaces, complete spaces, compact spaces, Baire category theorem.
Measure Spaces:
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.
Lp-Spaces:
Minkowski and Holder inequalities, Riesz representation theorem.
Banach Spaces:
Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem.
Hilbert Spaces:
Geometrical aspects, projection, Riesz-Fischer theorem.
Suggested Courses:
- MATH 62051 / 72051:Functions of a Real Variable I
- MATH 62052 / 72052: Functions of a Real Variable II
Suggested References:
- H. L. Royden, Real Analysis, MacMillan
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill