What is $ i^i$, where $i$ is the imaginary unit. Apparently wolfram alpha and google give: $$i^i\approx0.207879576=e^{-\pi/2}$$ But how?    Maybe let me try: $$x=i^i\implies x=\exp(i\ln i)$$ $$x=\exp\left(i\left(i-\frac{i^2}2+\frac{i^3}3-\frac{i^4}4+\cdots\right)\right)$$ $$x=\exp\left(i\left(i-\frac{1}2-\frac{i}3+\frac{1}4+\cdots\right)\right)$$ $$x=\exp\left(-1-\frac{i}2+\frac{1}3+\frac{i}4+\cdots\right)=^{\text {?}}\exp(-\pi/2)$$
According to WolframAlpha,  but I don't know how I can prove it.
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