Representing $(\sin\pi t)$with Euler's formula

Question

So I know how to represent $\sin(kw_{0}t)$ using Euler's formula as $\frac{1}{2j}(e^{jkw_0t} - e^{-jkw_0t})$, but when I try to represent $sin(\pi t)$ using the same method I get: \begin{align*} \sin(\pi t) = \frac{1}{2j}(e^{j \pi t} + e^{-j \pi t}) \end{align*} My question is why can't I write $e^{j \pi t}$ as $(e^{j\pi})^t = (-1)^{t}$ and get $0$ for every $t$?

Answer 1

When working with complex number you have that $e^{i\pi t} \neq (e^{i\pi})^t$ because the complex logarithm is a multi-valued function, you can see it on Wikipedia here.