§ 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 .