# 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.)

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