On the nonoscillatory phase function for Legendre's differential equation

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. Our asymptotic expansion is not as efficient as the well-known uniform asymptotic expansion of Olver; however, unlike Olver's expansion, it coefficients can be easily obtained. Numerical experiments demonstrating the properties of our asymptotic expansion are presented.