This exam will test your working knowledge of basic real, complex and functional analysis. You will be required to demonstrate an ability to use standard results and techniques to solve problems, including special cases of standard theorems which do not require long arguments.

We will not emphasize the memorization of statements of theorems nor of long proofs of standard theorems.

The student is urged to work on the problems in the relevant sections and chapters in the reference books.

The syllabus is divided into the topics of **Complex Analysis** and **Real and Functional Analysis**.

1. Definition of Holomorphic fuctions with examples, including logarithms, roots, and Möbius transformations.

2. Cauchy-Riemann Equations.

3. Power Series Expansion and applications including the Identity Theorem.

4. Cauchy Integral Theorem.

5. Applications of Cauchy Integral Theorem to evaluating Riemann integrals and summation of infinite series, Rouché's Theorem, The Argument Principle, Open Mapping Theorem, Liouville's Theorem, The Fundamental Theorem of Algebra.

6. The Maximum Modulus Theorem and Applications including the Schwarz Lemma.

7. Limit properties of Holomorphic functions including Hurwitz' Theorem.

**References**

[2] Conway, Functions of One Complex Variable, Chapters 1,3,4,5,6.

[3] Schaum's Outline of Complex Variables, Chapters 3-8.

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Convergence a.e., convergence in measure, convergence in the mean and how they are related to each other, Egorov's Theorem, Luzin's Theorem.

Lebesgue integrals, Fatou's Lemma, Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem, differentiation across the integral sign; comparison of Riemann and Lebesgue integrals; sequence of functions with equi-absolutely continuous integrals: Vitali's Theorem on a set with finite measure ([1, pp 151-159], [4, pp 143, Exercise 10], equi-absolutely continuous integrals = uniformly continuous integrals).

Set of discontinuous points and differentiabilty of a monotone function, functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus and its counter-example (Cantor's function). ([1, pp 204-220], [3, Chapt. 5].) Lebesgue's density theorem.

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Abstract measure spaces, completion of measure spaces, one measure is orthogonal or absolute continuous with respect to another measure, Lebesgue decomposition, Radon-Nikodyn theorem; product measures, Fubini-Toneli Theorem.

Space of continuous functions, Weierstrass Theorem, Arzela-Ascoli Theorem ([3, Chapt. 9, Sec. 6, 7]).

Contraction Mapping Theorem on complete metric spaces with applications to initial value problems of ODE and integral equations ([2, Sec. 8], [5, Sec. 3.8]).

Hilbert spaces, Schwarz's inequality, orthogonal projection of a point onto a closed subspace, Riesz Representation Theorem ([4, pp 79-85], [5, Secs 6.1 and 6.2]). Bessel's inequality, Parseval's identity, Gram-Schmidt procedure, completeness of orthonormal sets, least-square approximation.

Normed vector spaces, Banach spaces, bounded operators, standard examples:

**References**

[2] Kolmogorov and Fomin, Introductory Real Analysis.

[3] Royden, Real Analysis

[4] Rudin, Real and Complex Analysis, 2nd edition.

[5] Friedman, Foundations of Modern Analysis.

[6] Folland, Real Analysis.

[7] Wheeden and Zygmund, Measure and Integral, an introduction to real analysis.

[8] Stein and Sharkarchi, Real Analysis, Measure Theory, Integration and Hilbert Spaces.

[9] Stein and Sharkarchi, Fourier Analysis.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu