I know where the $\frac{1}{2\pi}$ factor is from.

For light,

$c=\lambda.f$

For photon in circular motion,

$v=c=r\omega=r.2\pi.f$

so,

$\lambda = 2\pi .r$

which is very odd indeed, because $\lambda$ is straight $2\pi.r$ raps around the straight line.  It is possible because the two paths do not start at the same point and do not end at the same point.  In fact the two paths do not cross ever.  It is possible when

$r=\cfrac{\lambda}{2\pi}$

This relationship between $r$ and $\lambda$ makes it easy to obtain $r$.  All calculations involving $r$ can be replaced with $\frac{\lambda}{2\pi}$.

$\hslash$, and life goes on.