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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 ( n-1)(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 1-z = 1 z ² 2 z 1-2 z ³ + 1 1-z = 3-4 z (1-z )(1-2 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 z-e − z ) / 2, get e z =-i and compute the logarithm.)...

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- Spring '14
- Sidorov
- Equations, Quadratic equation, Elementary algebra