[Math] Asymptotic Approximation of Spherical Bessel Function for Large Arguments

asymptoticsbessel functions

I'm looking for the asymptotic approximation to the spherical Bessel function of the first kind $j_n (\rho)$ for $\rho\gg n^2$. This is mainly to speed up calculation of a light propagation problem. There are examples for spherical Hankel functions but I've not been able to find an equivalent for $j_n (\rho)$. Does anyone know of any examples of this?

Best Answer

The spherical Bessel function is defined as $j_{n}(z)=\sqrt{\tfrac{1}{2}\pi/z}J_{n+\frac{1}{2}}\left(z \right).$ To get an asymptotic approximation for $j_n,$ use the Hankel expansion http://dlmf.nist.gov/10.17.E3: $$J_{\nu}(z) \sim \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left( \cos\omega\sum_{k=0}^{\infty} (-1)^{k}\frac{a_{2k}(\nu)}{z^{2k}}-\sin\omega\sum_{k=0}^{\infty}(-1)^{k} \frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),$$ with $\omega=z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi,$ and $$a_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-1)^{2})}{k!8^{k}} \;\cdot$$