Is $\sqrt{-1}$ equal to $i$ or $\pm i$?

Question

In complex numbers is $\sqrt{-1}$ equal to $i$ or $\pm i$ ?

In both cases how do we explain it?

The question arose when I saw it in Lathi's book (Linear Systems and Signals).

Answer 1

$i^2=-1$ and $(-i)^2=-1$. In $\mathbb C$ these are the two distinct roots of equation $z^2+1=0$.

Based on that it might be tempting to say that $\sqrt{-1}=i$ and/or $\sqrt{-1}=-i$, but we deal here with a function on $[0,\infty)\subset \mathbb R$ so $\sqrt{-1}$ is not defined properly here. Extension of the function to $\mathbb C$ leads to 'branches'.