Real Exam Topics---Masters Orals
241-242.
Metric space topology: open, closed, compact, bounded, etc. Sequential compactness.
R^n: Heine-Borel, Bolzano-Weierstrass, Lindeloef. Counterexamples to Heine-Borel in other metric spaces.
Continuity: in metric spaces and R^n, especially R.
Uniform continuity.
Compactness + continuity implies uniform continuity. Compactness + continuity implies: boundedness, extrema are attained, compactness of range. Know some counterexamples.
Riemann integral. Definition (upper and lower sums). Sketch: f continuous on [a,b] implies \int_a^b f dx exists.
Sequences and series. Definitions related to convergence: convergence, Cauchy, bounded. Definition of convergent series. Absolute convergence. Tests for convergence: integral, comparison, root, ratio. Counterexamples for root and ratio (inconclusive cases).
Sequences and series of functions. Uniform convergence. Uniform convergence + continuity implies limit is continuous: know how to prove it! Know counterexample for non-uniformly convergent sequence. Weierstrass M-test. Radii of convergence for real power series.
Derivatives: Mean Value Theorem, Intermediate Value Theorem for derivatives.
333.
Sigma algebra. Measure space. Examples. Outer measure. Def of measurability.
Def of integral of non-negative simple functions and of non-negative measurable functions. Integrable functions.
Monotone Convergence, Fatou’s Lemma, Dominated Convergence; Egorov’s Theorem.
Density Theorems: approximability by continuous functions, step functions, simple functions, of functions in L^1(R). Definition of L^p.
Riemann-Lebesgue Lemma; Borel-Cantelli.
Fubini and Tonelli Theorems.
Complex Exam Topics---Masters Orals
Definition of analytic (complex differentiable). Cauchy-Riemann equations: show that f’(z) exists implies f=u+iv satisfies the CREs.
Line (contour) integrals: how defined. Be able to do a simple one.
Sketch proof of Goursat’s Theorem; indicate how it leads to Cauchy’s Theorem and the Cauchy Integral Formula on a disk.
Show how Cauchy Integral Formula leads to power series.
Show: if a power series at a converges at z, it converges at all z’ which are closer to a than z is.
Cauchy’s estimates for terms of a power series.
Liouville’s Theorem. Fundamental Theorem of Algebra.
Isolated zeroes. Orders of zeroes: definition and significance.
Singularities: removable, pole, essential. Know what they mean and relate them to Laurent series. Casorati-Weierstrass. Orders of poles: definition and significance.
Residues. Definition; how to compute in simple examples. Using residues to find definite integrals.
Simple conformal mappings: half-planes, sectors, infinite strips, half-disk ONTO the unit disk.