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Unformatted text preview: Solution. From the lectures, R = lim n → + ∞ ± ± ± ± a n − 1 a n ± ± ± ± = lim n → + ∞ (2 n − 1 + 1) n ( n1)(2 n + 1) = 1 2 . [2 marks] • Find a closed formula for f ′ ( z ) (without inﬁnite summation) for any z such that  z  < R . Solution. We have for  z  < 1 / 2, f ′ ( z ) = ∞ ∑ n =1 (2 n + 1) z n − 1 = 1 z ∞ ∑ n =1 2 n z n + ∞ ∑ n =1 z n − 1 = 1 z ( ∞ ∑ n =1 (2 z ) n ) + 1 1z = 1 z ² 2 z 12 z ³ + 1 1z = 34 z (1z )(12 z ) . [3 marks] Problem 4 Find all complex solutions z of the equation sinh z =i. Solution. We have sin( iz ) = i sinh z , whence sinh z =i sin( iz ). Therefore, sin( iz ) = 1, whence iz = π/ 2 + 2 πk, k ∈ Z . Answer: z =iπ/ 2 + 2 iπk, k ∈ Z . [5 marks] (Alternatively, one can write sinh z = ( e ze − z ) / 2, get e z =i and compute the logarithm.)...
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 Spring '14
 Sidorov
 Equations, Quadratic equation, Elementary algebra

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