Complex power of a real number

Question

What is the meaning of $(-1)^{i}$, where $i^{2}=-1$ and what is its value?

Answer 1

$$-1=e^{(2n+1)i\pi}\implies (-1)^i=e^{-(2n+1)\pi}$$ where $n$ is any integer

Answer 2

$(-1)^i$ is defined as $e^{i\log(-1)}$, where $\log(z)$ is a multivalued function, defined as the set of $w$ for which $e^w=z$. Since $e^w=-1$ exactly when $w=(2n+1)\pi i$, for integer $n$, we have that $$ (-1)^i=\{e^{i\cdot((2n+1)\pi i)}:n\in\mathbb{N}\}=\{e^{-(2n+1)\pi}:n\in\mathbb{N}\} $$