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!
Best Answer
Alternatively: If $z=a+bi$, then $\displaystyle a=\frac{z+\overline z}2$ and $\displaystyle b=\frac{z-\overline z}{2i}$. Since $\overline{z^z}=\overline z^{\overline z}$, the real and imaginary parts of $z^z$ are $\displaystyle \frac{z^z+\overline z^{\overline z}}2$ and $\displaystyle \frac{z^z-\overline z^{\overline z}}{2i}$.