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# ans9 - Math 185 Sample Answers to Problem Set#9 Page 214 8...

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Math 185. Sample Answers to Problem Set #9 Page 214 8. Let X n =0 a n ( z - z 0 ) n be the Taylor series for f ( x ) about z = z 0 , and let R be its radius of convergence. The assumption that f ( n ) ( z 0 ) = 0 for n m implies that a 0 = a 1 = · · · = a m = 0 , so f ( z ) = X n = m +1 a n ( z - z 0 ) n = ( z - z 0 ) m +1 X n =0 a n + m +1 ( z - z 0 ) n for all z with | z - z 0 | < R . (Here the last equality holds because the factor ( z - z 0 ) m +1 is just a constant as far as the summation is concerned—see Exercise 7 on page 181.) Therefore g ( z ) = X n =0 a n + m +1 ( z - z 0 ) n for all z satisfying 0 < | z - z 0 | < R . This is also true when z = z 0 since g ( z 0 ) = a m +1 = f ( m +1) ( z 0 ) ( m + 1)! . Therefore g ( z ) has a Taylor series at z 0 , so it is analytic there. 10. For all N Z + and all z in the given annular domain, let ρ N ( z ) = S 2 ( z ) - N - 1 X n =0 b n ( z - z 0 ) n , so that, as with (3) on page 207 (since 1 / ( z - z 0 ) n is also continuous on C ), Z C g ( z ) S 2 ( z ) dz = N - 1 X n =0 b n Z C g ( z )( z - z 0 ) - n dz + Z C g ( z ) ρ N ( z ) dz . Next, we need to show that the given power series is uniformly convergent on C (and this is the main difficulty in this part of the problem). Assume that our annular domain is R 1 < | z - z 0 | < R 2 . Then the power series T ( w ) = X n =1 b n w n 1

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2 is convergent on the set | w | < 1 /R 1 since T (1 / ( z - z 0 )) = S 2 ( z ) , term for term. If the contour C is given by z = z ( t ) , a t b , then we can let R 0 be the minimum
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ans9 - Math 185 Sample Answers to Problem Set#9 Page 214 8...

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