Let z₁=x₁+iy₁ or r₁e^(iθ₁),
where x₁=r₁cos(θ₁), y₁=r₁sin(θ₁) and e^(iθ₁)=cos(θ₁)+isin(θ₁),
θ₁=arctan(y₁/x₁), r₁=√(x₁²+y₁²).
Let z₂=x₂+iy₂ or r₂e^(iθ₂),
where x₂=r₂cos(θ₂), y₂=r₂sin(θ₂) and e^(iθ₂)=cos(θ₂)+isin(θ₂).
θ₂=arctan(y₂/x₂), r₂=√(x₂²+y₂²).
z₁^z₂=(x₁+iy₁)^(x₂+iy₂)=[(r₁e^(iθ₁))^x₂][(r₁e^(iθ₁))^(iy₂)].
(r₁e^(iθ₁))^x₂=(r₁^x₂)e^(ix₂θ₁)=(r₁^x₂)(cos(x₂θ₁)+isin(x₂θ₁)).
(r₁e^(iθ₁))^(iy₂)=(r₁^(iy₂))(e^(-y₂θ₁)).
r₁^(iy₂)=e^(iy₂ln(r₁))=cos(y₂ln(r₁))+isin(y₂ln(r₁)).
Therefore:
z₁^z₂=(r₁^x₂)(cos(x₂θ₁)+isin(x₂θ₁))(cos(y₂ln(r₁))+isin(y₂ln(r₁)))(e^(-y₂θ₁)),
z₁^z₂=(r₁^x₂)e^(-y₂θ₁)(cos(x₂θ₁+y₂ln(r₁))+isin(x₂θ₁+y₂ln(r₁)).
EXAMPLES
(1) z₁=1, z₂=i, so x₁=y₂=1, x₂=y₁=0, θ₁=0, θ₂=π/2, r₁=r₂=1.
z₁^z₂=(1)(1)(1)=1, so 1^i=1.
(2) z₁=1+i, z₂=1-i, so x₁=x₂=y₁=1, y₂=-1, θ₁=π/4, θ₂=-π/4, r₁=r₂=√2.
z₁^z₂=2.8079+1.3179i.
|z₁^z₂|=3.1018 approx.