# On the nonoscillatory phase function for Legendre's differential equation

## TL;DR

This paper introduces a nonoscillatory phase function for Legendre's differential equation, enabling more stable and efficient evaluation of Legendre functions of large orders through asymptotic expansion and numerical validation.

## Contribution

It presents the first derivation of a nonoscillatory phase function for Legendre's equation and develops an asymptotic expansion for large-order Legendre functions.

## Key findings

- The phase function is nonoscillatory and well-behaved.
- The asymptotic expansion improves evaluation accuracy for large orders.
- Numerical experiments confirm the effectiveness of the proposed methods.

## 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.