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For fixed order, the standard Hankel function expansions in $e^{\pm iz}{z}^{-1/2}$ are valid in different sectors on a chosen argument sheet. Analytic continuation can mix exponentials; the sectors must be specified rather than treating a formal inverse-power series as global.

In conformal Newtonian gauge, a free scalar has no linear anisotropic stress. Eliminating its perturbation with the momentum constraint gives ${\Phi }_k^{\prime \prime }+3{\Phi }_k^{\prime }/\tau +k^2{\Phi }_k=0$. Substitution of ${\Phi }_k=(-\tau {)}^{-1}y(-k\tau )$ gives the order-one Bessel equation. The displayed Hankel function has positive incoming frequency $e^{-ik\tau }$ at early times.

With constant permeability, the electric equation reduces to $({\partial }_x^2+{\partial }_z^2+k^2(z))E_y=0$. Insert the oblique-wave dependence $E_y=e_y(z)e^{-i{k}_0\sin {\theta }_{0}x}$ and set $s=z-z_0$. The inverse-square refractive-index profile givesSubstituting $e_y=\sqrt{|s|}w(\beta |s|)$ gives the Bessel equation of order $\nu$. On either side of the profile singularity, when ${\beta }^2>0$,Equivalently use Hankel functions for an incoming/outgoing wave basis. If ${\beta }^2<0$, write $\gamma =\sqrt{-{\beta }^2}$ and replace $J_{\nu },Y_{\nu }$ by modified Bessel functions $I_{\nu },K_{\nu }$ with argument $\gamma |s|$. For $\beta =0$, the solutions are $|s{|}^{1/2+\nu }$ and $|s{|}^{1/2-\nu }$, with a logarithmic second solution in the repeated-root case $\nu =0$. The constants on each side require incident-wave and boundary or matching data; the supplied singular profile alone does not choose them.

Take the logarithmic derivative of the proposed asymptotic expansion. Through order $z^{-1}$, $u^{\prime }/u=\lambda +\mu /z+O(z^{-2})$ and $u^{\prime \prime }/u={\lambda }^2+2\lambda \mu /z+O(z^{-2})$. Matching the constant and inverse-power terms givesThe second formula assumes a simple root, $2\lambda +f_0\ne 0$. At a repeated root the ansatz can fail: the next equation first requires $f_1\lambda +g_1=0$, and if that holds higher orders determine the remaining behaviour.

For the Bessel differential equation, $\lambda =\pm i$ and $\mu =-1/2$. Set $u=e^{\lambda z}{z}^{-1/2}v(z)$. Direct substitution leaves $v^{\prime \prime }+2\lambda v^{\prime }+(1/4-{\nu }^2)z^{-2}v=0$. With $v=1+a/z+O(z^{-2})$, matching $z^{-2}$ gives $a=(1/4-{\nu }^2)/(2\lambda )$. ThusTwo constant multiples of the Hankel functions normalize their leading coefficients to one and realize these solutions in their appropriate sectors.

For generic $\nu$, these are sectorial expansions, not unchanged expansions for a single branch over arbitrarily continued $\arg z$. The factors $z^{-1/2}$ require a branch, and crossing sector boundaries can introduce the other exponential by Stokes phenomenon and analytic continuation. More precisely, the standard normalized first Hankel solution has its expansion for $-\pi +\delta \le \arg z\le 2\pi -\delta$, and the second for $-2\pi +\delta \le \arg z\le \pi -\delta$, with any fixed positive $\delta$ inside these sectors. In particular both are valid in the common slit sector $-\pi <\arg z<\pi$, away from its boundaries. These sector statements are recorded in NIST's Hankel asymptotics. Exceptional half-integer orders have terminating elementary expressions, so a blanket assertion of nonzero Stokes mixing for every order would be too strong.

Outside a short-range scattering region, the stationary Schrödinger equation reduces to the two-dimensional Helmholtz equation. Separation in polar coordinates gives outgoing radial Hankel functions, whose large-$r$ dependence is $r^{-1/2}{e}^{ikr}$. The same power follows from conserved probability current: a circle has length proportional to $r$, so the squared outgoing amplitude must decrease as $1/r$.

For an incident plane wave in the $x$ direction, adopt the dimensionless amplitude conventionThe phase $e^{i\pi /4}$ and factor $k^{-1/2}$ specify the definition of $f$; they can be absorbed into a differently normalized scattering amplitude. Here they are kept exactly so that the later two-dimensional scattering cross-section follows consistently.