Complex analysis: Evaluating an i^i

"Evaluate $\displaystyle i^{1+i}$. Put the result in the form $\displaystyle u+iv$ when $\displaystyle u,v$ are real."

$\displaystyle i = e^{i(\frac{\pi}{2}+2n\pi)}$

$\displaystyle i^{1+i} = [e^{i(\frac{\pi}{2}+2n\pi)}]^{1+i} = e^{-(2n+\frac{1}{2})\pi}e^{i(2n+\frac{1}{2})\pi}$

$\displaystyle = e^{-(2n+\frac{1}{2})\pi}[\cos(2n+\frac{1}{2})\pi + i\sin(2n+\frac{1}{2})\pi] = e^{-(2n+\frac{1}{2})\pi}$ for $\displaystyle n = 0, \pm1, \pm2,...$

Looks good or did I not need to start with that $\displaystyle 2n\pi$?