§ 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 .