118Homework6 - 3 iz 2 + 2 z-1 + i and that C is the circle...

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Homework 6 Due Monday, Oct. 11 1. Suppose that f is an entire function and let C be a simple closed contour oriented in the positive direction. Also let z 0 C . If z 0 is on the interior of C , then Cauchy’s Integral Formula implies that Z C f ( z ) z - z 0 dz = 2 πif ( z 0 ) . If z 0 is on the contour C , then the integral R C f ( z ) z - z 0 dz is undefined be- cause the integrand is not defined at a point on C . What is the value of R C f ( z ) z - z 0 dz when z 0 is outside of C ? 2. Evaluate the following integrals: (a) R C e z z - 2 dz , where C is the circle | z | = 3 oriented positively. (b) R C e z z - 2 dz , where C is the circle | z | = 1 oriented positively. (c) R C e 3 z z - πi dz , where C is the circle | z - 1 | = 4 oriented positively. (d) R C e iz z 3 dz , where C is the circle | z | = 2 oriented positively. (e) R C cos( πz ) z 2 - 1 dz , where C is the circle | z | = 2 oriented positively. 3. Suppose that a > 0 is a real number and that C is the circle | z | = 3 oriented in the positive direction. Prove that 1 2 πi Z C e az z 2 + 1 dz = sin a. 4. Evaluate 1 2 π R 2 π 0 sin 2 ( π/ 6 + 2 e ) . 5. Suppose that f ( z ) = z 5 -
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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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