The integral formulas of the associated Legendre functions

作     者：Yu, Jinhai Wan, Xiaoyun Zeng, Yanyan 

作者机构：Chinese Acad Sci Grad Univ Coll Earth Sci Beijing 100049 Peoples R China 

出 版 物：《JOURNAL OF GEODESY》 (大地测量学杂志)

年 卷 期：2012年第86卷第6期

页      面：467-473页

核心收录：

学科分类：070801[理学-固体地球物理学] 07[理学] 08[工学] 0708[理学-地球物理学] 0816[工学-测绘科学与技术] 

主　　题：Legendre function Integral formula Recursion method 

摘      要：A new kind of integral formulas for is derived from the addition theorem about the Legendre Functions when - is an even number. Based on the newly introduced integral formulas, the fully normalized associated Legendre functions can be directly computed without using any recursion methods that currently are often used in the computations. In addition, some arithmetic examples are computed with the increasing degree recursion and the integral methods introduced in the paper respectively, in order to compare the precisions and run-times of these two methods in computing the fully normalized associated Legendre functions. The results indicate that the precisions of the integral methods are almost consistent for variant in computing , i.e., the precisions are independent of the choice of on the interval [0,1]. In contrast, the precisions of the increasing degree recursion change with different values on the interval [0,1], particularly, when tends to 1, the errors of computing by the increasing degree recursion become unacceptable when the degree becomes larger and larger. On the other hand, the integral methods cost more run-time than the increasing degree recursion. Hence, it is suggested that combinations of the integral method and the increasing degree recursion can be adopted, that is, the integral methods can be used as a replacement for the recursive initials when the recursion method become divergent.