Abstract:In this paper, the associated Legendre function, with arbitrary order n and degree m Pmn(cosθ), is represented as the product sum of the coefficient E(k) and the angle (n-2k)θ′ss sine or cosine. The available value range of k is from 0 to int\[n/2\]. 〖JP2〗When the degree m is less than or equal to 2, E(k) can be expressed by the coefficient of P0n(cosθ) expansion. Otherwise, it will be a linear combination of several arrays. The given analytical expression not only helps to understand the Legendre function’s characteristics and property proof, but also can simplify its application in the related technical field.