Unformatted text preview: z , and Log z is one of them. All other values of log z diﬀer from Log z by an integer multiple of 2 πi . This is because the imaginary part of the logarithm is de±ned by using a branch of arg z , and the branches of arg z diﬀer by integer multiples of 2 π . (See Applied Complex Analysis and PDE for more details on the logarithm.) In particular, the imaginary part of Log z , which is Arg z , is in the interval (π, π ]. You can use Mathematica to evaluate Log z and e z . This is illustrated by the following exercises. 21. We have (1) · (1) = 1 but 0 = Log 1 ± = Log (1) + Log (1) = iπ + iπ = 2 iπ. 25. (a) By de±nition of the cosine, we have cos( ix ) = e i ( ix ) + ei ( ix ) 2 = ex + e x 2 = cosh x....
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 Fall '11
 StuartChalk
 Complex number, arg

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