Preliminary Exam in Applied Mathematics - Part B. Analysis

Preliminary Exam in Applied Mathematics - Part B. Analysis

Three courses in real and complex analysis are recommended for students preparing for Part B:

Material Covered

The real analysis portion of the exam focuses mainly on material covered in the first course and a portion of the second of the two-semester sequence Analysis I and Analysis II (Math 645 and Math 745). Source material for the exam can be found in the standard introductory graduate texts on real and functional analysis (a sample is listed below). A knowledge of advanced calculus is assumed as covered in the introductory analysis and advanced calculus courses Math 480–481 and Math 545–546, material that can be found in such texts as “Mathematical Analysis” by Apostol and “An Introduction to Analysis” by Wade.

Reference Texts

  • “Real Analysis” by Royden and Fitzpatrick
  • “Analysis” by Lieb and Loss
  • “Applied Analysis”, by Hunter and Nachtergaele
  • “A Course in Real Analysis” by McDonald and Weiss

Detailed Outline, Real Analysis

Point-set Topology and Metric Spaces

  • Open sets, closed sets, convergent sequences
  • Continuous mappings
  • Complete metric spaces
  • Compact metric spaces
  • Contraction mapping theorems

Measure Theory and Integration

  • Basic notions of measure theory 
  • Measurable functions and Lebesgue integral
  • Lebesgue measure, product measures, Fubini-Tonelli theorem
  • Lebesgue monotone and dominated convergence theorems, Fatou’s lemma
  • Approximations by continuous functions, convolutions, simple functions

Function Spaces

  • Lp spaces, normed spaces, dual spaces
  • Jensen, Minkowski, Hölder and Young inequalities 
  • Banach and Hilbert spaces, types of convergence

Harmonic Analysis

  • Fourier series and Fourier transform
  • Bases, orthogonality, completeness
  • Bessel’s inequality, Parseval’s theorem, Plancherel theorem

Detailed Outline, Complex Analysis

  • Function of a complex variable, its limits and continuity.
  • Complex differentiability and analyticity, the Cauchy-Riemann equations, harmonic functions
  • Multivalued functions, branch points and branch cuts, and basic Riemann surfaces
  • Contour integration of a function of complex variable
  • Integral theorems: the Cauchy-Goursat theorem, the Cauchy's Integral Formula and the Morera theorem
  • Corollaries of integral theorems: Liouville's theorem, the Maximum Modulus Principle and their implication
  • Complex series, in particular the Taylor and Laurent series, their uniqueness and domains of convergence
  • Classification of zeros, isolated and non-isolated singularities of a complex function; the little Picard theorem
  • The concept of analytic continuation, the Identity theorem, and the Schwartz reflection principle
  • Cauchy Residue theorem and its application to complex integration contours; the Jordan's lemma
  • Basic understanding of simple conformal mappings, for instance the inversion map.
  • The Argument Principle and the Rouche's theorem

Copies of past qualifying exams are available here.
Updated: December 7, 2017