Laguerre Polynomials

doi:10.1117/3.2207310.ch9

Book: Special Functions for Optical Science and Engineering

Author(s): Vasudevan Lakshminarayanan, L. Srinivasa Varadharajan

Published: 2015

https://doi.org/10.1117/3.2207310.ch9

Author Affiliations +

Abstract

The Laguerre functions (Edmond Laguerre, 1843–1886) are solutions to the differential equation, appropriately named the Laguerre differential equation, where n is a constant. If n ¼ 0,1,2,3,. . . , one of the solutions of Eq. (9.1) becomes a polynomial called the Laguerre polynomial. This solution is finite for all finite values of x and tends to infinity no faster than ex/2 as x!`. These polynomials arise in many areas of modern physics, most notably in quantum mechanics (the hydrogen atom) as well as in optics (fiber optics with a parabolic refractive index distribution).