Question about complex numbers (what's wrong with my reasoning)?

Question

Can someone point out the flaw here?

$$e^{-3\pi i/4} = e^{5\pi i/4}$$

So raising to $\frac{1}{2}$, we should get

$$e^{-3\pi i/8} = e^{5\pi i/8}$$

but this is false.

Answer 1

Paraphrase using $e^0=1$ and $e^{\pi i}=-1$. We can write $$ e^{-3\pi i/4}\;1^2=e^{-3\pi i/4}\;(-1)^2 $$ Raising to the $\frac12$ power yields $$ e^{-3\pi i/8}\;1=e^{-3\pi i/8}\;(-1) $$ The problem is that without proper restrictions (e.g. branch cuts), the square root is not well-defined on $\mathbb{C}$.

Answer 2

The problem is that $(e^x)^y=e^{xy}$ does not hold with complex numbers as it does with real numbers. This can change what the principal value, which is what has happened in your example. You should read a bit about principal logarithms and branch cuts.