MA530_JAN10 - z ) g ( z ) is real for all z D . Prove that...

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MATH 530 Qualifying exam, January 2010, A. Eremenko Notation: < z and = z are real and imaginary parts of z , C is the complex plane, “entire” means analytic in C , “meromorphic function” means a ratio of two analytic functions. Each problem is worth 10 points. 1. Find a conformal map f of the upper half-plane onto the region { z C : | z - 2 i | < 2 , | z - i | > 1 } such that f (0) = 2 i,f ( ± 1) = 0. 2. Let f be an analytic function in the closed unit disc which satisfies | f (0) | + | f 0 (0) | < inf {| f ( z ) | : | z | = 1 } . Prove that f has at least two zeros (counting multiplicity) in the open unit disc. 3. Let f and g be two entire functions, and | f ( z ) | ≤ | g ( z ) | for all complex z . Prove that f = cg , where c is a constant, or g = 0. 4. Let f be an analytic function in the unit disc satisfying |< f ( z ) | < 1 for all z in the unit disc. Prove that | f 0 (0) | ≤ 2. 5. Let f and g be two analytic functions in a region D . Suppose that f (
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Unformatted text preview: z ) g ( z ) is real for all z D . Prove that f = cg where c is a real constant, or g . 6. Let f be an analytic function in the region { z : 0 &lt; | z | &lt; 1 } , and suppose that it satises | f ( z ) | A + B | z |- for some positive numbers A,B and . Prove that f is meromorphic in the unit disc (that is 0 is either a pole or a removable singularity of f ). 7. Let f = X n =1 a n z n be a conformal map from the unit disc onto the square { z : |&lt; z | &lt; 1 , |= z | &lt; 1 } . Prove that a n = 0 for all n 6 = 4 k + 1 , k = 0 , 1 , 2 , 1 .... 1 8. Evaluate the integral Z log x 1 + x 2 . Hint: integrate log z/ (1 + z 2 ) over the boundary of the region { z : &lt; | z | &lt; R, = z &gt; } . 2...
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MA530_JAN10 - z ) g ( z ) is real for all z D . Prove that...

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