How do $i$ and $\pi$ show up in this result from a Google search?

Question

I am just curious and play a bit here. I googled (-1e-10)^(-1e-10) and got
(-1e-10)^(-1e-10) = $1 - 3.14159266 × 10^{-10} i$

I guess this is some form of approximation to $x^x$ when $x$ approaches $0$. But what does $i$ stand for and how does $\pi$ show up in this?

Note that googling 1e-10^1e-10 simply returns 0.99999999769.

Answer 1

If $r$ is a small positive real, one branch of exponentiation gives$$(-r)^{-r}=(re^{i\pi})^{-r}=\underbrace{r^{-r}}_{\approx1}\underbrace{e^{-ir\pi}}_{\approx1-ir\pi}\approx1-ir\pi.$$