On the nonoscillatory phase function for Legendre's differential equation
Provides a new computational tool for evaluating Legendre functions of large orders, relevant to applications in physics and engineering.
The authors derive a nonoscillatory phase function for Legendre's differential equation, enabling efficient evaluation of Legendre functions of large orders via an asymptotic expansion. Numerical experiments confirm the expansion's accuracy.
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.