Besselfunctionintroduce
Bessel functions are a general term for a special type of mathematical function. The general Bessel function is
The standard solution function y(x) of the following ordinary differential equations (generally called Bessel equations):
The solutions to such equations cannot be expressed systematically by elementary functions.
The specific form of the Bessel function changes with any real number α in the above equation (correspondingly, α is called its corresponding Bessel
the order of the function). The most common case in practical applications is that α is an integer n, and the corresponding solution is called the n-order Bessel function.
Although in the above differential equation, the positive and negative signs of α itself do not change the form of the equation, it is still accustomed to target α and
−α defines two different Bessel functions (this can bring benefits, such as eliminating the non-smoothing function at point α=0
sex).
history
Several positive integer order special cases of Bessel function were studied by Swiss mathematician Daniel Bernoulli as early as the mid-18th century
The chain was proposed when vibrating, which aroused interest in the mathematics community at that time. Daniel's uncle Jacob Bernoulli, Euler, Lagrand
Mathematicians such as Japan have made important contributions to the study of Bessel's functions. In 1817, German mathematician Bessel started research
When Puller proposed the motion problem of the three-body gravity system, the overall theoretical framework of Bessel function was systematically proposed for the first time.
Later generations named this function after him [1] [2].
Realistic background and scope of application
Bessel equation is obtained when using the separation variable method to solve Laplace equation and Helmholtz equation under cylindrical or spherical coordinates.
What we have achieved (the whole order form α = n in the cylindrical domain problem; the semi-odd order form is obtained in the spherical domain problem
α = n+½), so the Bessel function plays a very important role in fluctuation problems and various problems involving potential fields.
The most typical problems are:
● Electromagnetic wave propagation problems in cylindrical waveguides;
● Heat conduction problems in cylinders;
● Vibration mode analysis problem of circular (or ring) film;
In some other fields, Bessel functions are also quite useful. For example, FM synthesis in signal processing
Or the definition of Kaiser window, the Bessel function must be used.
Definition The Bessel equation is a second-order ordinary differential equation, and there must be two linearly irrelevant solutions. For various specific situations
Moreover, different forms of expressing these solutions are proposed. The following are the different types of Bessel functions.
The first Bessel function
Figure 2 The first Bezier function (Betzel J function) curves of order 0, order 1 and order 2
(As shown below, the first type of Bessel function is sometimes referred to as "J function", please pay attention to the readers.)
The first type of α-order Bessel function J (x) is the solution of the Bessel equation when α is an integer or α is non-negative, and must be satisfied when x=0
α
limited. The reasons for selecting and processing J in this way are shown in the properties below this topic; another way of defining it is through it in x=
α
Taylor series expansion of point 0 (or more generally expansion by power series, which applies to α being non-integral):
In the above formula, Γ(z) is the Γ function (it can be regarded as the generalization of factorial functions to non-integral independent variables). The first type of Bessel function
The shape is roughly similar to the sine or cosine function at a rate decay (see the introduction to their progressive forms below this page
), but their zero points are not periodic, and as x increases, the interval between zero points will become closer and closer to periodicity.
Figure 2 shows the curve of the first Bessel function J (x) of the 0-order, 1-order and 2-order (α = 0,1,2).
α
If α is not an integer, then J (x) and J (x) are linearly independent, and can form a solution system of differential equations. Otherwise, if α
−
α α
It is an integer, so the above two functions satisfy the following relationship:
Therefore, the linear irrelevant conditions are no longer met between the two functions. In order to find another that is linearly independent of J (x) in this case
α
In one solution, you need to define the second type of Bessel function, and the definition process will be given in the following subsections.
Bessel points
Another way to define Bessel function when α is an integer is given by the following integral:
(See page 360 of the expression when α is any real number)
This integral formula is the definition proposed by Bessel back then, and he also deduces some properties of the function from this definition. another
The integral expression is:
Relationship with super-geometric series
The Bessel function can be represented by supergeometric series as follows:
The second type of Bessel function (Neuman function)
Figure 3 Curve diagram of the second Bezier function (Betzer Y function) in order 0, order 1 and order 2
(As shown below, the second type of Bessel function is sometimes referred to as "Y function", please pay attention to the reader.)
The second type of Bessel function may be more commonly used than the first type. This function is usually represented by Y (x), they are Bessel square
α
Another type of solution to the process. The point x = 0 is the (infinite) singularity of the second type of Bessel function.
Y (x) is also called Neumannfunction,