Category Archives: Special Functions

Neumann’s Formula

Today I learned a pretty cool formula called Neumann’s formula while reading a paper by Maurice Lévy, Wave equations in momentum space, Proceedings of the Royal Society of London. Series A, Vol. 204, No. 10 (7 December 1950), pp. 145-169. When $n$ is a positive integer and $|z|>1$, the Legendre function of the second kind $Q_n(z)$ can be expressed in terms of the Legendre function of the first kind $P_n(x)$ as
A derivation of this formula can be found on p. 320 of E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th edition, Cambridge University Press, 1927 as cited in the Lévy’s paper (the page number is incorrectly cited as p. 330). The formula is originally appeared in a paper by J. Neumann (the author’s name is incorrectly cited as F. Neumann in Whittaker & Watson), Entwicklung der in elliptischen Coordinaten ausgedrückten reciproken Entfernung zweier Puncte in Reihen, welche nach den Lalace’schen $Y^{(n)}$ fortschreiten; und Anwendung dieser Reihen zur Bestimmung des magetischen Zustandes eines Rotations-Ellipsoïds, welcher durch vertheilende Kräfte erregt ist. pp. 21-50 (the formula appears on page 22), Journal für die reine und angewandte Mathematik (Crelle’s journal), de Gruyter, 1848. The title is unusually long. It reads like an abstract rather than a title. Maybe it was not unusual back then.

The formula can be used to evaluate the following integral
$$I=2\pi\int_0^\pi\frac{P_l(\cos\theta)\sin\theta d\theta}{|\vec{p}-\vec{p’}|^2+\mu^2}$$
which appears in the momentum representation of Schrödinger equation with Yukawa potential. With $x=\cos\theta$, the integral $I$ can be written as
\begin{align*} I&=2\pi\int_{-1}^1\frac{P_l(x)dx}{p^2+p’^2-2pp’x+\mu^2}\\ &=\frac{2\pi}{pp’}\frac{1}{2}\int_{-1}^1\frac{P_l(x)dx}{\frac{p^2+p’^2+\mu^2}{2pp’}-x}\\ &=\frac{2\pi}{pp’}Q_l\left(\frac{p^2+p’^2+\mu^2}{2pp’}\right) \end{align*}