homework108c2

homework108c2 - F ( z ) = ( f ( z ) if | z | = 1 1 2 i H f...

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HOMEWORK 2 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, APRIL 14, 2009 (1) Calculate I ∂D (2 , 4) ζ ( ζ + 3) ( ζ + i )( ζ - 8) dζ, where the circle is traversed once in positive direction. (2) Let a > 0 and b R , and choose ω [0 ,π/ 2) with cos( ω ) = a/ a 2 + b 2 . Calculate Z 0 e - ax cos( bx ) dx by considering the (complex) contour 0 R Re 0, consisting of two straight lines from the origin and one arc of the circle with radius R around the origin. (3) Let f be a continuous function on ∂D (0 , 1) = { z C | | z | = 1 } . Define
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Unformatted text preview: F ( z ) = ( f ( z ) if | z | = 1 1 2 i H f ( ) -z d if | z | < 1 where is a curve traversing the unit circle once in positive direction. Is F continuous on D (0 , 1) = { z C | | z | 1 } ? (4) Let f be holomorphic on D (0 , 1) = { z | | z | < 1 } and continuous on D (0 , 1). Moreover suppose | f ( z ) | 1 if | z | = 1. Prove that | f ( z ) | 1 for all z D (0 , 1). 1...
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