COMBINED ANALYSIS

This exam involves combining Complex Analysis with Real Analysis into one exam format. The following are the materials you will need to review for this exam:

COMPLEX 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 62151/72151-2 
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-Stieljes 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-Weierstras 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 62051/72051-2 
Suggested References: 
· H. L. Royden, Real Analysis, MacMillan. 
· W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.