I use the standard notations. When $x$ is real then by definition $ I_{\nu}(x)=e^{-\nu\pi i/2}J_{\nu}(ix). $ I want to define $I_{\nu}$ for complex $z$. Watson (Treatise of the Theory of Bessel Functions, 2nd ed, 3.7, p.77) says it is $ \begin{cases} I_{\nu}(z)=e^{-\nu\pi i/2}J_{\nu}\left(ze^{\pi i/2}\right), & (-\pi<\arg z\leq\pi/2),\\ I_{\nu}(z)=e^{(3/2)\nu\pi i}J_{\nu}\left(ze^{-(3/2)\pi i}\right), & (\pi/2<\arg z\leq\pi). \end{cases} $
Q1. $e^{\pi i/2}\neq e^{-(3/2)\pi i}$?
Q2. Why not good to simply put $z$ in place of $x$?
Q3. How was chosen these two cases of argument of $z$? What is the reason?
Watson says "it is usually convenient", but I want mathematical justification.