Math 536 FINAL PREPARATION Spring 2008
THEORETICAL QUESTIONS
(1) State and sketch the proof of Runge’s Theorem.
(2) State and sketch the proof of Mittag-Leffler Theorem.
(3) State the Weierstrass Product Theorem and sketch the proof in the case when the
domain is the entire complex plane.
(4) Give the definition of the Gamma Function and state its basic properties (meromorphic
continuation, zeros, poles, functional equation, product formula).
(5) Give the definition of the Riemann Zeta Function and state a few of its properties (at
least, Euler’s product formula, meromorphic continuation, poles, and what is known
about the zeros).
(6) Give the definition of a subharmonic function on a domain D⊂Cand sketch the proof
of the Strict Maximum Principle.
(7) State and sketch the proof of Harnack’s Principle (about a sequence of positive harmonic
functions on a planar domain).
(8) Describe the Perron procedure for solving the Dirichlet problem on a planar domain.
(9) Give the definition of subharmonic barrier and state the consequences of its existence.
Give an explicit example of a subharmonic barrier.
(10) Give the definition of the Green’s function for planar domains with piecewise analytic
boundary and state its basic properties.
(11) Give the definition of the Green’s function for general planar domains and state its basic
properties.
(12) Give the definition of a Riemann surface and give an example (e.g. the torus or the
Riemann surface of log z) explicitly defining the coordinate maps.
(13) Give the definition of the Green’s function on a Riemann surface (include the definition
of a Perron family). When does the Green’s function exist?
(14) Define the analytic continuation along a path and state the Monodromy Theorem (either
in the complex plane or on a Riemann surface; it’s your choice).
(15) State the Uniformization Theorem and indicate the main steps of the proof in the case
when the Green’s function exists.
(16) Give the definition of a covering map, the universal covering map, and covering trans-
formations, and state their basic properties.
