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Unformatted text preview: z , and Log z is one of them. All other values of log z dier from Log z by an integer multiple of 2 i . This is because the imaginary part of the logarithm is dened by using a branch of arg z , and the branches of arg z dier by integer multiples of 2 . (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 (-, ]. You can use Mathematica to evaluate Log z and e z . This is illus-trated by the following exercises. 21. We have (-1) (-1) = 1 but 0 = Log 1 = Log (-1) + Log (-1) = i + i = 2 i. 25. (a) By denition of the cosine, we have cos( ix ) = e i ( ix ) + e-i ( ix ) 2 = e-x + e x 2 = cosh x....
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- Fall '11