I have some trouble understanding how to pass from bessel function to spherical bessel function. Departing from the Helmholtz differential equation:
$ \begin{equation} r^2 \frac{\partial^2 R}{\partial r^2} +2r \frac{\partial R}{\partial r} +\left[ k^2 r^2 - m(m+1)\right]R =0\text{.} \label{eq:zap1} \end{equation}$
Then using the change of function $R(r)=\frac{Z(r)}{(kr)^{1/2}} $, it's possible to find : $Z" r^{2} + Z' r +\left[ k^2 r^2 - (m+1/2)^2\right]Z=0$.
Thus the solution using the bessel equation is : $R(r)= A\frac{J_{m+1/2}(kr)}{\sqrt{kr}}+B\frac{Y_{m+1/2}(kr)}{\sqrt{kr}}$.
The general solution is known as the spherical bessel function : $R(r)= A'j_{m}(kr)+B'y_{m}(kr)$,
avec :
$j_{m}(z)=\sqrt{\frac{\pi}{2}}\frac{J_{m+1/2}(z)}{\sqrt{z}} $ et $y_{m}(z)= \sqrt{\frac{\pi}{2}}\frac{Y_{m+1/2}(z)}{\sqrt{z}} $.
So where this $\sqrt{\frac{\pi}{2}}$ come from ?
(The details of the equations are in http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html)