One solution to the bessel equation
$$\left[x^2\partial_x^2 + x \partial_x + x^2 - \alpha^2\right] y(x) = 0$$
are the Hankel functions $H^{(1)}_\alpha(x)$ and $H^{(2)}_\alpha(x)$. I am interested in their asymptotic limit for complex numbers $z$ with $|z|\rightarrow \infty$. According to Wikipedia the leading terms are then
What I do not understand is the range of the arguments of the complex number $z$. Why is it necessary to define these limits for a range $-2\pi < \arg(z) < 2\pi$? Naively I would assume that $\exp(i\pi) = \exp(-i\pi)$, why is the distinction still important?