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Principal Branch and Complex Powers

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How do you find (and what is it) the principal branch of the following complex powers:

(e/2)(1-i(sqrt(3)))^(3pi*i)

and

(1-i)^(4i).

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Solution Summary

Principal branch and complex powers are investigated. The solution is detailed and well presented.

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1st question:

First recall that

(1-i*sqrt(3))/2 = cos(pi/3) - i*sin(pi/3) = cos(-pi/3) + i*sin(-pi/3) = e^{-i*pi/3+2i*pi*n},

where n can take values n = 0, +1, -1, +2, -2, ......

Each n is regarded as a "branch" and the principal branch is n= 0, that is we take ...

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