Negative imaginary exponents

Question

I was reading this question earlier: Understanding imaginary exponents

In the answer, the answerer says

$A^i=x+iy$

Furthermore, we can write $A^{−i}=x−iy$ for the same $x$ and $y$.

Can someone explain to me why this is true? It kind of makes sense to me but how can we make the leap to just negating the imaginary component?

Answer 1

Hint: $$A^i=x+iy$$ $$A^{-i}=\frac{1}{A^{i}}=\frac{1}{x+iy}=\frac{1}{x+iy}\cdot\frac{x-iy}{x-iy}=\frac{x-iy}{x^2+y^2}=x-iy$$

Note : it says $$|A^i|=1$$

$$|A^i|=\sqrt{x^2+y^2}=1$$ $$\Rightarrow x^2+y^2=1$$

Answer 2

If $A>0$ is real, we define $A^z=e^{z\ln A}$. So

$A^i=e^{i\ln A}=\cos(\ln A)+i\sin(\ln A)$ $A^{-i}=e^{-i\ln A}=\cos(-\ln A)+i\sin(-\ln A)=\cos(-\ln A)-i\sin(\ln A)$