# 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 R^{n}, 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.

###### L_{p}-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