On the nonoscillatory phase function for Legendre's differential equation
Numerical Analysis
2017-10-11 v1 Classical Analysis and ODEs
Abstract
We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic derivative of this solution, we show that Legendre's differential equation admits a nonoscillatory phase function. Moreover, we derive from our expression an asymptotic expansion useful for evaluating Legendre functions of the first and second kinds of large orders, as well as the derivative of the nonoscillatory phase function. Numerical experiments demonstrating the properties of our asymptotic expansion are presented.
Cite
@article{arxiv.1701.03958,
title = {On the nonoscillatory phase function for Legendre's differential equation},
author = {James Bremer and Vladimir Rokhlin},
journal= {arXiv preprint arXiv:1701.03958},
year = {2017}
}