§ 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 .

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 .