This is a follow up to this question.
How does one arrive at the asymptotic expressions for the bessel functions?
After looking at: G. N. Watson, "A Treatise on the Theory of Bessel Functions", 2nd Edition, 1948.
I arrived at the solution, however, I noticed that the results for finite real z will be complex for Iv. I understand that in the limit the complex part will be zero but I'm just confused as to it's existence in the first place. For integer $\nu$ it is either +i or -i.
$$ I_v(z) = \frac{1}{(2\pi z)^{1/2}} e^{z} \sum_{m=0}^\infty \frac{(-1)^m(\nu,m)}{(2z)^m} + \frac{1}{(2\pi z)^{1/2}} e^{-z+(\nu+1/2)\pi i} \sum_{m=0}^\infty \frac{(\nu,m)}{(2z)^m} $$
As an asymptotic series for $z \to +\infty$ the terms with the imaginary component will not contribute since they are exponentially small compared to the terms in the first series. However, as $z \to -\infty$ the terms of the first series are exponentially small compared to those of the second series, so only the terms in the second series contribute.
For $\nu = 1/2$, for instance, $I_\nu(z)$ is complex for $z < 0$, and the second series captures the asymptotic behavior as $z \to -\infty$.