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# COMPLEX ANALYSIS MATH 536

UW
GPA 3.76

Boris Solomyak

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COURSE
PROF.
Boris Solomyak
TYPE
Study Guide
PAGES
2
WORDS
KARMA
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## Popular in Mathematics (M)

This 2 page Study Guide was uploaded by Addison Beer on Wednesday September 9, 2015. The Study Guide belongs to MATH 536 at University of Washington taught by Boris Solomyak in Fall. Since its upload, it has received 33 views. For similar materials see /class/192065/math-536-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15
Math 536 FINAL PREPARATION Spring 2008 THEORETICAL QUESTIONS 1 State and sketch the proof of Runge7s Theorem 2 State and sketch the proof of Mittag Lef er 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 de nition of the Gamma Function and state its basic properties meromorphic continuation7 zeros7 poles7 functional equation7 product formula 5 Give the de nition of the Riemann Zeta Function and state a few of its properties at least7 Euler7s product formula7 meromorphic continuation7 poles7 and what is known about the zeros 6 Give the de nition of a subharmonic function on a domain D C C and sketch the proof of the Strict Maximum Principle 7 State and sketch the proof of Harnack7s Principle about a sequence of positive harmonic functions on a planar domain A 00 V Describe the Perron procedure for solving the Dirichlet problem on a planar domain A K V Give the de nition of subharmonic barrier and state the consequences of its existence Give an explicit example of a subharmonic barrier 10 Give the de nition of the Greens function for planar domains with piecewise analytic boundary and state its basic properties 11 Give the de nition of the Greens function for general planar domains and state its basic properties 12 Give the de nition of a Riemann surface and give an example eg the torus or the Riemann surface of log 2 explicitly de ning the coordinate maps 13 Give the de nition of the Greens function on a Riemann surface include the de nition of a Perron family When does the Greens function exist 14 De ne the analytic continuation along a path and state the Monodromy Theorem either in the complex plane or on a Riemann surface its your choice 15 State the Uniformization Theorem and indicate the main steps of the proof in the case when the Greens function exists 16 Give the de nition of a covering map7 the universal covering map7 and covering trans formations7 and state their basic properties PROBLEMS 1 Let R llll 2397 where H is the upper half plane Find the value gR2 3239 of the Greens function at z 2239 with pole at q 3239 2 Let S be the strip S z39y 0 lt y lt 1 and let A be the annulus A 2 1lt lt 2 Find explicitly a covering map 7139 S a A and determine all covering transformations 3 Let R be a Riemann surface and suppose that there is a non constant bounded analytic function b R a C Show that R possesses a Green7s function 4 Show that 00 f8 z z p n1 is analytic in U gt 1 and that 8 expf8 5 Prove that isolated singularities are removable for bounded harmonic functions 6 Let u be subharmonic in D Show that 27r ureitdt 0 is increasing as a function of r 0 S r lt 1 7 Using basic principles eg considerations of uniform convergence7 periodicity and Liouville7s Theorem7 show that 7139 1 sin2 7T2 7 200 2 7 71 39 8 Write down an in nite product that converges to an entire function fz with zeros of order 1 at the points 2 W n 2 17 and no other zeros Prove the convergence of your product

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