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exam_4_m3364_summer_2011_v1_0

# exam_4_m3364_summer_2011_v1_0 - f z centered at z = 0...

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Exam 4: Math 3364 Summer 2011 Problem 1. (10) Find the radius of convergence of the power series deﬁned by X n =1 ( - 1) n 3 n n 5 ( z - i ) n . Problem 2. (5 each, 10 total) Let f be the rational function deﬁned by f ( z ) = 1 1 - z . (a) Find a general formula for f ( n ) ( z 0 ) (the n th derivative of f at z 0 ). (b) Show that the Taylor series for f centered at z 0 = i is given by f ( z ) = X n =0 ( z - i ) n (1 - i ) n +1 . Problem 3. (10) Let g be a real-valued continuous function deﬁned on the real interval [0 , 1] and for | z | < 1 deﬁne H ( z ) = Z 1 0 g ( t ) 1 - zt 2 d t. Find the Taylor series for H centered at z 0 = 0. Hint: think about geometric series. Problem 4. (5 each, 10 total) For each of the following statements, determine if the statement is true or false. (a) If f ( z ) is analytic on all of the complex plane, then the Taylor series for
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Unformatted text preview: f ( z ) centered at z = 0 converges to f ( z ) for every z in C . (b) Let Γ be a simple closed positively oriented contour. Suppose that f ( z ) is analytic everywhere on and inside Γ except at a single point z 1 inside Γ. If z 1 is a pole of order 2, then Z Γ f ( z ) d z 6 = 0 . Problem 5. (10) Find the Laurent expansion of the function f ( z ) = 1 / ( z + z 2 ) that is valid in the annulus < | z + 1 | < 1. Problem 6. (10) Evaluate I | z | =1 e z ( z-2 i )( z-3) d z. Problem 7. (10) Evaluate I | z | =3 e z z ( z-2) 2 d z. 1...
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