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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!

Jyrki Lahtonen
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Ahna Akbar
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  • Related : http://math.stackexchange.com/questions/9776/how-to-raise-a-complex-number-to-the-power-of-another-complex-number and http://math.stackexchange.com/questions/201991/for-what-values-alpha-for-complex-z-lnz-alpha-alpha-lnz?lq=1 – lab bhattacharjee Sep 19 '15 at 06:04

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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 $\displaystyle \frac{z^z-\overline z^{\overline z}}{2i}$.

Christopher Carl Heckman
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