Complex argument imaginary number

Could anyone clarify this for me?

Wolfram:
http://www.wolframalpha.com/input/?i=arg%28i%5Ei%29

But as per my derivation:
$[i^i]=[e^{iln(i)}] = i\frac{\pi}{2}+2\pi k$, since $ln(i) = i\frac{\pi}{2}$

What am I doing wrong?

Answer

As you say, $\ln(i)=i\frac \pi 2$. When you multiply that by $i$, as you show in $e^{i\ln(i)}$, you get $-\frac \pi 2$. Then you add in the $2\pi k$ just fine.