Let lambda_1 = 1. lambda_2 = i, lambda_3 = - 1 and lambda_4 = -i. Find a domain
ID: 1720445 • Letter: L
Question
Let lambda_1 = 1. lambda_2 = i, lambda_3 = - 1 and lambda_4 = -i. Find a domain D containing the points lambda_1, lambda_2, lambda_3, lambda_4 and an analytic function f on D such that e^f(lambda_l) = lambda_1, e^f(lambda_2) = lambda_2, and e^f(lambda_1) = lambda_4. where e^z is the complex exponential function. Calculate explicitly the values f(lambda_1).f(lambda_2),f(lambda_3) and f(lambda_4), and also find the value of the derivative f' at 1. As in part a), hut now find a domain D and an analytic function f on D such that [f(lambda_1)]^3 = lambda_1]^3 = lambda_1, [f(lambda_2)]^3 = lambda_2, [f(lambda_3)]^3 = lambda_3 and [f(lambda_4)]^3 = lambda_4, Also find f(lambda_2) and f'(lambda_3).Explanation / Answer
In both cases , take the domain to be D = Complex plane from which the half-line y=x starting from the origin and lying in the positive quadrant is removed.
As this domain is simply connected , we can find a branch of log which is analytic and use this to contruct the functions satisfying the conditions of a) and b)
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