§ 4 Legendre function


First,        the definition of Legendre function


[ Legendre functions of the first kind ]


It resolves single-valued in the removed plane .

[ Legendre functions of the second kind ]


It resolves single-valued in the removed plane .





It resolves single-valued in the removed plane .



    [ General Legendre function ]



They are single-valued analytically in the removed plane and are Legendre differential equations


two linearly independent solutions of .

At that time , they were Legendre functions of the first and second kinds, respectively .

when a positive integer), there are





for having



( At that time , the Legendre polynomial









2.        Other expressions of Legendre function







where is a forward simple closed curve on the plane (Fig. 12.2 ), the enclosing point is the sum , but not the enclosing point .

     When (or when an integer),       










The integral route is shown in Figure 12.3. At that time ,








3.        The recurrence formula and related formulas of the Legendre function









The above formula is also applicable to , just replace P in the formula with . Use


The corresponding recursive formula on the interval can be obtained , and there are similar formulas for .


4.        Orthogonality of Legendre functions


Only the orthogonality of the function is a positive integer, and the formula is as follows






5.        Asymptotic expressions and inequalities of Legendre functions


[ asymptotic expression ]





[ inequality ]





The inequalities are real numbers and positive integers .


Original text