Lommel Function

There are several functions called ``Lommel functions.'' One type of Lommel function is the solution to the Lommel Differential Equation with a Plus Sign,

$$y=ks_{\mu,\nu}(z),$$ (1)

where

$$s_{\mu,\nu}^{(+)}(z)\equiv{\textstyle{1\over 2}}\pi \left[{Y... ...^\mu J_\nu(z)\,dz-J_\nu(z)\int_0^z z^\mu Y_\nu(z)\,dz}\right].$$ (2)

Here, $J_\nu(z)$ and $Y_\nu(z)$ are Bessel Functions of the First and Second Kinds (Watson 1966, p. 346). If a minus sign precedes $k$, then the solution is

$$s_{\mu,\nu}^{(-)} \equiv I_\nu(z)\int_z^{c_1} z^\mu K_\nu(z)\,dz-J_\nu(z)\int_{c_2}^z z^\mu I_\nu(z)\,dz,$$ (3)

where $K_\nu(z)$ and $I_\nu(z)$ are Modified Bessel Functions of the First and Second Kinds.

Lommel functions of two variables are related to the Bessel Function of the First Kind and arise in the theory of diffraction and, in particular, Mie scattering (Watson 1966, p. 537),

$$U_n(w,z) = \sum_{m=0}^\infty (-1)^m\left({w\over z}\right)^{n+2m}J_{n+2m}(z)$$ (4)

$$V_n(w,z) = \sum_{m=0}^\infty (-1)^m\left({w\over z}\right)^{-n-2m}J_{-n-2m}(z).$$ (5)

See also Lommel Differential Equation, Lommel Polynomial

References

Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 369, 1960.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.