Show the limiting forms of the Bessel functions

Question

If we have the Bessel functions of the first kind, $J_{\nu}(x)$ and we consider two cases where $x\ll1$ and $x\gg1,\nu$. I often see that for $x\ll1$

$J_{\nu}(x)\rightarrow\frac{1}{\Gamma(\nu+1)}(\frac{x}{2})^{\nu}$

and then for $x\gg1,\nu$

$J_{\nu}(x)\rightarrow\sqrt{\frac{2}{\pi x}} \cos(x - \frac{\nu \pi}{2} - \frac{\pi}{4})$

The two terms shown are the leading terms only. My question is, does anyone know of a proof for this? Or can some one explain to me how to show these limiting forms for the Bessel function of the first kind.

Thanks.

Answer 1

$J_\nu(x)$ is an entire function, hence the asymptotics for $x\ll 1$ just comes from the Taylor series at $x=0$. The asymptotics for $x\gg 1$ come from the application of the saddle point method to the integral representation for $J_n(x)$: $$ J_n(x) = \frac{1}{2\pi i}\oint e^{(z/2)(t-1/t)}\frac{dt}{t^{n+1}}. $$ A very clear complete derivation can be found here.