complex analysis - Find the real and Imaginary part of $z^{z}$.

Find the real and Imaginary part of $z^{z}$.

My approach: If $z=re^{i\theta}$, then $$z^{z}=\exp{(z\ln(z))}=\exp{(re^{i\theta}(\ln(r)+i(\theta+2k\pi))}$$
$$=\exp{(r(\cos(\theta)+i\sin(\theta))(\ln(r)+i(\theta+2k\pi)))}$$
$$=\exp(r(\cos(\theta)\ln(r)-\sin(\theta)(\theta+2k\pi))+ir(\cos(\theta+2k\pi)+\sin(\theta)\ln(r)))$$

And continuous with this development, I can find Imaginary and real part, but is this correct?? Exist any approach more easy?? Regards!