how to raise a complex number to the power of another

Question

I have to raise z1 to the power of z2, where both z1 and z2 are complex numbers. How should i proceed?

Answer 1

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.