Value of $(-1)^x$ for $x$ irrational

I was working on an analysis problem when this question arose in one my proofs. I think it may be either $-1$ or $1$, but it seems like there can only be an arbitrary way to assign this.

So is there an agreed upon method for determining $(-1)^x$ for irrational values of $x$?

The general idea is that $a^b$ is to be defined as $\exp(b\log a)$, where that’s the natural logarithm in the parentheses. The problem is that the log has no unambiguous definition for numbers that are not positive real; generally there’s an ambiguity of multiples of $2\pi i$. Since “the” log of $-1$ is $(2k+1)\pi i$ for all integers $k$, you get infinitely many values for $(-1)^x$ when $x$ is not a rational number.

Euler's Formula ($e^{ix}=\cos(x)+i\sin(x)$) is pretty useful here. Since $\cos((2n+1)\pi)=-1$ and $\sin((2n+1)\pi)=0$ for integer $n$, $$\cos((2n+1)\pi)+i\sin((2n+1)\pi)=e^{i(2n+1)\pi}=-1$$

$$ (-1)^z = (e^{i(2n+1)\pi})^z = \cos((2n+1)\pi z)+i\sin((2n+1)\pi z) \tag{$n\in\mathbb{Z}$} $$ (This equation can also be viewed as a Generalization of De Moivre's formula)

Since $n$ goes over all integers, $(-1)^z$ has infinite values if the exponent $z$ is irrational. If $z$ is rational, its values repeat after a certain value of $n$.

Take $(-1)^\pi$ as an example: $$ \begin{align} (-1)^\pi &= (e^{i(2n+1)\pi})^\pi\\ &=e^{i(2n+1)\pi^2} \\ &=\cos((2n+1)\pi^2)+i\sin((2n+1)\pi^2) \tag{$n\in\mathbb{Z}$} \end{align} $$

As others have pointed out, using $a^x=\exp(\ln(a)x)$ to define exponentials with irrational exponents, the expression $(-1)^x$ would designate $\exp(m\pi\mathbf ix)$ for some odd integer $m$ depending on how the natural logarithm of $-1$ is chosen, which can be done in infinitely many ways. What I want to add to other answers is that, while some value for $m$ can be chosen, the fact that this ambiguity exists removes all utility that using the expression $(-1)^x$ might be imagined to have.

The situation is not at all like that of for instance the complex logarithm or arctangent functions, which functions cannot be continuously defined everywhere in the complex plane except at their singularities, which forces making branch cuts and choosing a value somewhere in the complement of the cuts to determine the branch. In those cases there is an inevitable problem in passing from a locally well defined function a globally defined one, the different branches share all the essential properties required (being local inverses of the exponential respectively tangent function, having a specific derivative) and can be transformed into one another by the process of gradually moving branch cuts. For $(-1)^x$ none of this applies. Each individual function $x\mapsto \exp(m\pi\mathbf ix)$ is defined without problem on the whole complex plane, and they have quite different properties; if one did not know that they were obtained as candidates for representing $x\mapsto(-1)^x$, it would be hard to say what property they have in common that sets them apart from other (exponential) functions. So if one wants to make one specific choice for $m$, say $m=1$, then by all means use $\exp(\pi\mathbf ix)$ and write it like that; however nothing is gained (except a great potential for confusion) by calling this $(-1)^x$. And by the above discussion, the term Principal Value is misplaced.

You could think in this: $$(-1)^x=e^{i\pi x}$$

After a brief commentary actually one really have is: $$-1=e^{i(\pi +2k\pi)}, k \in \mathbb{Z}$$ And then: $$(-1)^x=e^{i(\pi +2k\pi)x}, k \in \mathbb{Z} $$ And one would have more than one value for $(-1)^x$