Does any theory of infinite quantities provide info for $-1$ raised to infinite power, its absolute value, finite part and a series or integral form?

Question

I've heard about surreal numbers, hyperreal numbers, Hardy fields, nonstandard analysis, cardinal ariththmetic, ordinal arithmetic, games, etc.

My impression is that neither of them can exactly show the result of the following operation: $(-1)^w$, where $w$ is some infinite number from that theory.

The only theory that exactly gives a result is extended real line ${\overline {\mathbb {R} }}$, where, of course, $(-1)^\infty=0$ (because of Cesàro mean).

Am I correct in my skepticism about all these theories?

Answer 1

In the context of hyperreal numbers (defined using ultrapowers, for instance), $(-1)^w$ is defined for any infinite hypernatural number $w$. Specifically, $(-1)^w=1$ if $w$ is even and $(-1)^w=-1$ if $w$ is odd. More generally, $(-1)^w$ can be defined as a hypercomplex number for any hyperreal $w$ in the same way as it is defined for reals, as $\exp(i\pi w)$. (Of course, just as for real numbers, this is only one possible "branch" of the exponential; you could similarly say $(-1)^w$ is $\exp((i\pi+2ki\pi)w)$ for any hyperinteger $k$.)

I would also mention that $(-1)^\infty$ is usually considered to be undefined in the context of the extended real numbers. There may be contexts where it is convenient to define it to be $0$, but this is certainly not a widespread convention.

Answer 2

There are somewhat natural sine and cosine functions on the class $\mathbf{No}$ of surreal numbers. The class of zeroes of $\sin$ is $\pi \ \mathbf{Oz}$ where $\mathbf{Oz}$ is Conway's ring of omnific integers (a class of "integers" for $\mathbf{No}$).

For $a \in \mathbb{No}$, one can define $(-1)^a:=\cos(a \pi)+i\sin(a \pi) \in \mathbf{No}[i]$ (algebraic closure), which yields $(-1)^{a+b}=(-1)^a (-1)^b$ for all $a,b \in \mathbf{No}$. Note that for $z \in \mathbf{Oz}$, we have $(-1)^z=1$ if $z$ is even and $(-1)^z=-1$ if $z$ is odd.

Now I don't claim that this definition is useful or interesting. Is it the right definition of $(-1)^a$? To answer that, one should first say what one expects of such an operation...