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Unformatted text preview: 3 iz 2 + 2 z-1 + i and that C is the circle | z | = 2 oriented in the positive direction. (a) Prove that all the zeros of f ( z ) lie inside of C . (b) Calculate the value of R C f ( z ) f ( z ) dz . 6. Prove that the zeros of z 5 + z-16 i all lie between the circles | z | = 1 and | z | = 2. 7. Suppose that f ( z ) and g ( z ) are analytic inside and on a simple closed curve C except that f ( z ) has zeros z 1 , . . . , z j and poles w 1 , . . . , w k of orders n 1 , . . . , n j and p 1 , . . . , p k respectively. Prove that 1 2 i Z C g ( z ) f ( z ) f ( z ) dz = j X r =1 n r g ( z r )-k X r =1 p r g ( w r ) . (Hint: Mimic the proof of the Argument Principle and use Cauchys Inte-gral Formula to evaluate the integrals that appear.)...
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- Fall '09