This exam involves combining Complex Analysis with Real Analysis into one exam format.


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.)


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

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.


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