# Final Exam Solutions on Complex Variables and Applications...

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Final ExamSpring 2013Math 1560100 points totalS o l u t i o n s1. Find all the values of (-i)i. Find Im((-i)i).
2. Prove the trigonometric identity sin2z=1-cos 2z2by using the definition of trigonometricfunctions in terms of exponential functions.2-4=1-e2iz+e-2iz
3. Letf(z) = (z2+ 4)1/2. Show thatz= 2iis a branch point off(z).
4. For each limit determine if it exists. If the limit does not exists then prove it. If the limitexists then evaluate it.(a)limzeπi/2z2z2+ 1(z-eπi/2)
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(c) limz0(cosz)1/z2