Hankel Functions

Examining the asymptotic forms, we see that two particular complex linear combinations of the stationary solution have the behavior, at infinity, of an outgoing or incoming spherical wave when the time dependence is restored:

$$h^+_\ell(x) = j_\ell(x) + i n_\ell(x) \quad (= h^1_\ell(x))$$ (11.95)

$$h^-_\ell(x) = j_\ell(x) - i n_\ell(x) \quad (= h^2_\ell(x))$$ (11.96)

the spherical hankel functions of the first ($+$) (outgoing) and second ($-$) (incoming) kinds. Both of these solutions are singular at the origin like $1/x^{\ell+1}$ (why?) and behave like

$$\lim_{ x \rightarrow \infty} h^+_\ell(x) = (-i)^{\ell+1} \frac{e^{ix}}{x}$$ (11.97)

$$\lim_{ x \rightarrow \infty} h^-_\ell(x) = (i)^{\ell+1} \frac{e^{-ix}}{x}$$ (11.98)

at infinity. Two particularly useful spherical hankel functions to know are the zeroth order ones:

$$h^+_0(x) = \frac{e^{ix}}{ix}$$ (11.99)

$$h^-_0(x) = \frac{e^{-ix}}{-ix}$$ (11.100)