Syllabus for the Analysis Qualifying exam

 

The analysis qualifying exam covers the following five topics that roughly correspond to seven Yale Analysis classes.  You can use the descriptions below as well as the references for the relevant classes (available via Canvas) to prepare.

 

 

Properties of real numbers, limits, convergence of sequences and series, power series, Taylor series, differentiation and integration, metric spaces, Lebesgue integration, Fourier series, and applications to differential equations.  See for example [R, Chapters 1, 7, and 9] and [E, Appendix C].

 

Differentiability of complex functions, complex integration and Cauchy’s theorem, series expansions, calculus of residues, conformal mapping, Rouché’s theorem, Hurwitz theorem, Runge’s theorem, analytic continuation, Schwarz reflection principle, Jensen’s formula, infinite products, Weierstrass theorem. Functions of finite order, Hadamard’s theorem, meromorphic functions, Mittag-Leffler’s theorem, and subharmonic functions.  See for example [R, Chapters 10-16].

 

 

Construction and limit theorems for measures and integrals on general spaces, product measures, Lp spaces, and integral representation of linear functionals.  See for example [R, Chapters 2, 6, 7, and 8] and [E, Appendix E].

 

 

Hilbert, normed, and Banach spaces, geometry of Hilbert space, Riesz-Fischer theorem, dual space, Hahn-Banach theorem, Riesz representation theorems, linear operators, Baire category theorem, uniform boundedness, open mapping, closed graph, spectral theorem for compact self-adjoint operators, and Fredholm alternative.  See for example [R, Chapters 3, 4, and 5] and [E, Appendix D].

 

 

Wave equation, Laplace’s equation, heat equation, method of characteristics, and calculus of variations.  See for example [E, Chapters 2, 3 and 8].

 

[E] L. C. Evans, Partial Differential Equations, 2nd Edition, GSM 19, American Mathematical Society.

 

[R] W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill.