I have been asked to find the principal value of: $\big[\frac{e}{2}(-1-\sqrt3i)\big]^{3\pi i}$
The textbook Complex Variables with Applications by A. David Wunsch only provides answers to the odd questions and I have found myself at a loss. All examples I have found have been much simpler in nature. any help would be appreciated.
Best Answer
$1+i\sqrt 3 = 2 (\frac 12 + i \frac {\sqrt {3}}{2}) = 2(\cos \frac {\pi}{3} + i\sin \frac {\pi}{3}) = 2e^{\frac {\pi}{3}i}$
$\left((\frac {e}{2})(2e^{\frac {\pi}{3}i})\right)^{3\pi i}\\ \left(e^{\frac {\pi}{3}i+1}\right)^{3\pi i}\\ e^{(\frac {\pi}{3}i+1)\cdot (3\pi i)}\\ (e^{-\pi^2}e^{3\pi i})\\ e^{3\pi i} = -1\\ -e^{-\pi^2}\\ $