Asymptotics of Bessel functions

Remarks

The small-argument forms are the leading terms of the power series of $J_\nu$ and of the series of $Y_\nu$ obtained from its definition. The large-argument forms come from an integral representation of the Hankel functions $H^{(1)}_\nu = J_\nu + iY_\nu$ and the stationary-phase method. They are the first terms of full asymptotic expansions in powers of $\frac{1}{x}$. Example: for $\nu = \frac{1}{2}$, $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ exactly, and indeed $\cos\left( x - \frac{\pi}{2} \right) = \sin x$, with no error term. The large-argument form holds for fixed $\nu$; it fails when $\nu$ grows with $x$, for instance when $\nu$ is close to $x$, where the transition is described by Airy functions.

Going deeper

For a student

The large-argument form explains why the positive zeros of $J_\nu$ are eventually spaced by about $\pi$. The amplitude $\frac{1}{\sqrt{x}}$ reflects conservation of energy for cylindrical waves, whose wavefronts have length proportional to $x$. The small-argument forms decide which solution is allowed in a boundary value problem on a disc: regularity at the centre excludes $Y_\nu$. For modified Bessel functions the analogues are $I_\nu(x) \sim \frac{e^x}{\sqrt{2\pi x}}$ and $K_\nu(x) \sim \sqrt{\frac{\pi}{2x}}\, e^{-x}$.