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

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)ln(1−x) and ln(1+x)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].