§ 5   Bessel function

 

1.        Bessel functions of the first kind

 

[ Definition and Expression of Bessel Functions of the First Kind ]

   

is called a Bessel function of the first order, and it is single-valued in the plane except the semi-real axis (and when integer, in the full plane) . It satisfies the Bessel differential equation

                      

The constants (real or complex) in an equation are called the order of the equation or the order of the solution .

When (integer), is the generating function:

                =

and have

       

       

             

                

                                            

       

              

               

                                            

       

       

                     

              

       

       

    [ integral expression ]

               (Poisson integral representation)

                           (represented by Bessel integral)

                

            

                                                   

       

            

       

                              at the point,

The integral route is in the shape of ” as shown in the figure, at the point

       

       

   [ Related formula ]   

   

where are the two positive zeros of the function .

       

where are the two positive zeros of the function , and are any given constant .

            (addition formula)

       

where and represents the distance from the origin to any two points on the plane , and is the angle of intersection of the sum .

[ asymptotic expression ]

   

         

                                   fixed,

               fixed,

                                       

                             

                                     

                       (in

 

Second,        the second kind of Bessel function (Neumann function)

 

[ Definition and other expressions of Bessel functions of the second kind ]

   

It is called the Bessel function of the second kind ( also recorded in some books ), also known as the Neumann function, which is also the solution of the Bessel differential equation ( 1 ), where it is the Bessel function of the first kind ,

                

and single-valued analysis in the plane excluding the semi-real axis .

   

                  integer)

   

                 

                          is Euler's constant)

   

   

          

            

                                       

   

   

   

   

   

   

   

[ integral expression ]

         

   

        

           

[ asymptotic expression ]

   

          

                                  fixed,

   

                                      

 

Third,        the third kind of Bessel function (Hankel function)

 

[ Definition and Expression of Bessel Functions of the Third Kind ]

          

       

are called Bessel functions of the third kind, and Hankel functions of the first and second kinds, respectively, are single-valued analytically in the plane except the semi-real axis and satisfy the Bessel differential equation ( 1 ) .

                       

                       

       

              

              

       

              

              

       

       

                              

                             

    [ integral expression ]

       

       

       

       

             

           

       

       

             positive integer,

The integral route is shown in Figure 12.5.

[ asymptotic expression ]

   

                                        fixed,

   

                                        fixed,

   

   

                                      

                                       

 

Fourth,        the relationship between various Bessel functions and related formulas

 

[ Self-recursion relation ]   The following represents the Bessel function and .

   

   

           

   

   

   

[ Relationship between various Bessel functions ]

   

        

   

        

   

   

[ Other related formulas ]

    

   

   

   

   

   

 

5.        Variant Bessel function

 

[ Definition and Expression of Variant Bessel Function ]

   

   

                                  

Variant Bessel functions of the first and second kinds, also known as Basset functions, respectively, are single-valued in the plane with the semi-real axis removed .

                       

                           

   

                          ( as a positive integer)

   

                   

                        is Euler's constant)

   

       

                 

   

                       

   

   

   

   

[ integral expression ]

   

       

                                       

         

                                 

        

                                          

   

    is an integer)

[ Related formula ]

            

   

   

   

   

   

   

   

   

[ asymptotic expression ]

   

               fixed,

In the formula, the sign is selected as follows: at that time , take the positive sign, when,

Take a negative sign .

              

                            

                             

                                      

 

Original text