Complex number question......

What is 

\(i^i \) ?

Formula:

\(\begin{array}{|rcll|} \hline z &=& a+i\cdot b \\ \ln(z) &=& \ln(|z|) + i \cdot Arg(z) \qquad\qquad Arg(z) = \varphi \\ \ln(a+i\cdot b ) &=& \ln(\sqrt{a^2+b^2}) + i \cdot \arctan(\frac{b}{a}) \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline i^i &=& e^{i\cdot \ln(i) } \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline z &=& i \\ &=& 0+i\cdot 1 \qquad & | \qquad a = 0 \qquad b = 1\\ \ln(i) = \ln( 0+i\cdot 1 ) &=& \ln(\sqrt{0^2+1^2}) + i \cdot \arctan(\frac{1}{0}) \\ &=& \ln(1) + i \cdot \arctan(\frac{1}{0}) \qquad & | \qquad \ln(1) = 0\\ &=& 0 + i \cdot \arctan(\frac{1}{0}) \qquad & | \qquad \arctan(\frac{1}{0}) = \frac{\pi}{2} \\ &=& 0 + i \cdot \frac{\pi}{2} \\ &=& i \cdot \frac{\pi}{2} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline i^i &=& e^{i\cdot \ln(i) } \\ &=& e^{i\cdot i \cdot \frac{\pi}{2} } \\ &=& e^{i^2 \cdot \frac{\pi}{2} } \quad & | \quad i^2 = -1 \\ &=& e^{-\frac{\pi}{2} } \\ &=& 0.20787957635\dots \\ \hline \end{array}\)

\(\displaystyle {\imath^{\imath}=(e^{\imath\frac{\pi}{2}})^{\imath}=e^{-\frac{\pi}{2}}.}\)