MA530_AUG02 - f ( z ) on C-[0 , 1]. State Liouvilles...

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QUALIFYING EXAMINATION AUGUST 2002 MATH 530 - Prof. Bell 1. (20 pts) Suppose that f is a continuous complex valued function on the unit disc D 1 (0) and that f is analytic on the upper half disc, { z :Im z> 0 }∩ D 1 (0), and analytic on the lower half disc, { z :Im z< 0 }∩ D 1 (0). Use only Morera’s Theorem to prove that f is must actually be analytic on the whole disc. 2. (20 pts) Evaluate the integral Z 0 sin( x 2 ) dx by integrating e iz 2 around the con- tour that starts at the origin and follows the real line out to a point R> 0, then follows the circular arc Re it from t =0to t = π/ 4, and returns to the origin along the line joining Re iπ/ 4 to 0. Let R →∞ . You may use the fact that R 0 e - x 2 dx = π/ 2 without proving it. 3. (20 pts) Prove that the integral Z 1 0 t 2 t - z dt defines an analytic function
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Unformatted text preview: f ( z ) on C-[0 , 1]. State Liouvilles Theorem and use it to prove that f cannot be extended to [0 , 1] in such a way to make f an entire function. 4. (20 pts) a) (5 pts) Give a careful statement of the Schwarz Lemma. b) (15 pts) Prove that any analytic function f that maps the unit disc into itself, but is not one-to-one, must satisfy | f (0) | < 1. (Note, we do NOT assume that f (0) = 0 here.) 5. (20 pts) Suppose f is analytic on a neighborhood of the closed unit disc. If | f ( z ) | < 1 when | z | = 1, prove that there must exist at least one point z with | z | < 1 such that f ( z ) = z ....
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