Math3364-PracticeFinal-F10

Find all the points where f is analytic 6 10 points 5

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Unformatted text preview: . Consider f (z ) = e|z| . Find all the points where f is analytic. 6 (10 points) 5. Use Cauchy’s formula to prove that if f is entire, then for any R > 0, ￿ 2π 1 f (Reit )dt . f (0) = 2π 0 7 (10 points) 6. Calculate the Laurent series for the function f (z ) = √ annulus 2 < |z | < 2. 8 1 2+z 2 1 + 2−z and z in the (15 points) 7. Find all the solutions of the equation sin(z ) = cos(z ) and express them in terms of the (multi-valued) logarithm of suitable complex numbers. 9 [(15 points) 8.] Power and Taylor series (a) (5 points) Find the radius of convergence of the power series ∞ ￿1 zj 2 j j =1 (b) (10 points) Given that ∞ ￿ 1 = wj 1−w j =0 for |w| < 1, find the Maclaurin series fo...
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This note was uploaded on 01/21/2014 for the course MATH 3364 taught by Professor Staff during the Fall '08 term at University of Houston.

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