TITLE: Why does Wolfram Alpha say that $\infty^\infty$ results in "complex infinity"?
QUESTION [26 upvotes]: When entered into Wolfram Alpha, $\infty^\infty$ results in complex infinity.
Why is $\infty^\infty=\tilde\infty$?

REPLY [4 votes]: As far as I can tell: When $x$ and $y$ approach positive infinity, Wolfram Alpha assumes they may do so through the complex plane, as long as the arguments (angles) of $x$ and $y$ approach zero.
The argument of $x^{x+i}$ does not converge to any value as $x\to\infty$, even though the arguments of $x$ and $x+i$ go to zero. (This is because the argument is $\ln x$.) Thus, Wolfram Alpha responds with complex infinity (unknown argument) rather than positive infinity.