MA530_AUG94 - QUALIFYING EXAMINATION AUGUST 1994 MATH 530...

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Unformatted text preview: QUALIFYING EXAMINATION AUGUST 1994 MATH 530 All answers must be justified and work must be shown. 1. Let f be an analytic function in the open unit disk, |f (z )| ≤ 1, |z | < 1. Prove that |f (n) (0)| ≤ n!; n = 0, 1, 2, . . . . 2. Let f be a non-constant analytic function in a neighborhood N of the real axis R. Assume that Im f (z ) · Im z ≥ 0, z ∈ N. a) Show that f (z ) = 0, z ∈ R. b) Show that actually f (z ) > 0, 3. Evaluate the integral ∞ 0 z ∈ R. xα−1 dx , x+t where 0 < α < 1 and t > 0. 4. Find the one-to-one conformal map of the region {z : Re z > 0, Im z > 0, |z | > 1} onto the upper half-plane, such that i → 0, 1 → 1 and ∞ → ∞. 5. How many solutions (counting multiplicity) on the Riemann sphere can have the equation f ( z ) − z = 0, where f is a rational function of degree d ≥ 2. (The degree of f = P/Q is defined as max{deg P, deg Q}, where P and Q are polynomials without common factor.) 6. Describe the set in the complex plane where cos z is real. Draw the picture. 7. Find the residus of cot2 z at all isolated singular points. 1 ...
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This document was uploaded on 01/25/2012.

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