This-might-be-helpful-How-to-easily-calculate-z-1-z-2-with-z-1-z-2-C-Transform-z-1-re-i-and-z-2-a-bi-z-1-z-2-re-i-a-bi-z-1-z-2-r-a-bi-e-b-ia-z-1-z-2-r-a-e-

Question Number 192580 by Frix last updated on 21/May/23

This might be helpful :

How to easily calculate z_{1}^{z_{2}} with z_{1}, z_{2} \in C :

Transform z_{1} = r e^{i θ} and z_{2} = a + b i \Rightarrow

z_{1}^{z_{2}} = {(r e^{i θ})}^{a + b i}

z_{1}^{z_{2}} = r^{a + b i} e^{- b θ + i a θ}

z_{1}^{z_{2}} = r^{a} e^{- b θ} r^{b i} e^{i a θ}

z_{1}^{z_{2}} = \frac{r^{a}}{e^{b θ}} e^{i (a θ + b \ln r)}