CVEL8

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Evaluate (-i)^i\, on the principal branch.

(-i)^i=e^{i\mathrm{Log}(-i)}\, where capital functions are principal value.

And

\mathrm{Log}(-i) = \mathrm{Log}|-i| + i \mathrm{Arg}(-i)\,

= \mathrm{Log}(1) + i\frac{3\pi}{2}\,

= i\frac{3\pi}{2}\,

So

(-i)^i = e^{i\mathrm{Log}(-i)} = e^{i i\frac{3\pi}{2} } = e^{-\frac{3\pi}{2}} = e^{\frac{\pi}{2}}\,



Main Page : Complex Variables : Exponential and Log

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