Modified Bessel Equation

[thatboi]

Hey all,
I wanted to know if anyone knew somewhere I could find the asymptotic behavior for small x (i.e x approaching 0) limit of the modified Bessel equations with complex order. The wikipedia page for Bessel functions (https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_Iα,_Kα) provides a limit when $|z| \ll \sqrt{\alpha+1}$ but in the case of complex-valued $\alpha$ I am not sure how to interpret this bound.
Any assistance would be appreciated.

 

[Paul Colby]

They give

$I_\alpha(z) \approx \frac{1}{\Gamma(\alpha+1)}\left(\frac{z}{2}\right)^\alpha$

just treat this as an analytic function of $\alpha$ for small z.

 

[thatboi]

They give

$I_\alpha(z) \approx \frac{1}{\Gamma(\alpha+1)}\left(\frac{z}{2}\right)^\alpha$

just treat this as an analytic function of $\alpha$ for small z.

For the $K_{\alpha}$ limits, it is defined for $\alpha=0$ and $\alpha>0$, so for the case of complex $\alpha$, does this just mean $\alpha\neq 0$?

 

[Paul Colby]

That’s the way I read it. $z=0$ is a regular singular point and $\alpha$ is just a parameter. So it should be analytic.