814hw4sol - Math 814 HW 4 November 7, 2007 p. 74: 5, 6, 7,...

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Unformatted text preview: Math 814 HW 4 November 7, 2007 p. 74: 5, 6, 7, 9cd, 12, 13, 14. Exercise 5. Give the power series expansion of Log z about z = i and find its radius of convergence. For any nonzero a C , we have 1 z = 1 a 1 1 + z- a a = X n =0 (- 1) n a n +1 ( z- a ) n , with radius of convergence | a | . Take a = i , antidifferentiate, and remember that i 2 =- 1 , Log i = i/ 2 . You get Log z = i 2- X n =0 i n +1 n + 1 ( z- i ) n +1 = i 2- X n =1 ( iz + 1) n n , with radius of convergence | i | = 1 . Exercise 6. Give the power series expansion of z about z = 1 and find its radius of convergence. There are two branches of z , differing by a sign, which can be detected from the value 1 at z = 1 . Choose the branch f ( z ) such that f (1) = 1 . For n > , we have (CORRECTED VERSION) f ( n ) (1) = (- 1) n- 1 (2 n- 2)! 2 2 n- 1 ( n- 1)! . (Note: This is better than writing f ( n ) (1) = (- 1) n- 1 1 3 (2 n- 3) 2 n , 1 since the latter is ambiguous at n = 1 .) We get....
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814hw4sol - Math 814 HW 4 November 7, 2007 p. 74: 5, 6, 7,...

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