Unified solution for the Legendre equation in the interval [−1, 1]—An example of solving linear singular-ordinary differential equations

摘要

This study adopts the corrected Fourier series expansion method with only limited smooth degree to solve the Legendre equation with an arbitrary complex constant μ, and finds general solution for the intervals [0, 1] and [−1, 0], which includes a logarithm singular function in forms of ln(1−x) and ln(1+x), respectively, and a nonsingular function. The smooth conjunction of these two portions at x = 0 constructs the unified solution for the Legendre equation in the interval [−1, 1].