MA530_AUG08 - MATH 530 Qualifying Exam August 2008 (S....

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MATH 530 Qualifying Exam August 2008 (S. Bell, A. Eremenko) Each problem is worth 25 points 1. Suppose u and v are real valued harmonic functions on a domain Ω. a) (10 pts.) If u and v agree on a set with a limit point in Ω, does it follow that u = v on all of Ω? Explain. b) (15 pts.) If u and v satisfy the Cauchy-Riemann equations on a set with a limit point in Ω, does it follow that u + iv is analytic on Ω? Explain. 2. Suppose that f is continuous on { z : | z |≤ 1 } and analytic in { z : | z | < 1 } . a) (15 pts.) Prove that if f vanishes on { z : z = e , 0 θ π 2 } ,then f 0on the unit disc. Hint: Where do f ( i n z ) vanish for n =1 , 2 , 3? b) (10 pts.) Does the same result hold for harmonic functions? Explain. 3. Suppose f and g are analytic in a neighborhood of a point a , and suppose g has a simple zero at a . Find a formula for the residue of f ( z ) g ( z ) 2 at a in terms of f ( a ) and derivatives of f and g at a . 4.
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MA530_AUG08 - MATH 530 Qualifying Exam August 2008 (S....

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