Because special function is an important tool in mathematical analysis, the research and application of special function is very important. However, the special function is often not solved by one method, it is the flexible application of various methods and the concentrated expression of various thinking methods, so it is more difficult. The following is a sample essay on the properties and applications of several special functions that I have compiled. Welcome to reading.

The Properties and Applications of Several Special Functions

The special functions in mathematical analysis in this paper, such as gamma function, beta function, Bessel function and other hypergeometric sequence functions, have special properties and characteristics and have been widely used in reality. This paper briefly introduces the properties of the above three special functions and their applications in other fields, such as using special functions to find integrals and solving related physical problems. In this paper, firstly, the concepts and properties of several common special functions are summarized to deepen readers' understanding, and then relevant examples are analyzed in detail to achieve the purpose of flexible application.

Keywords special function; Nature; Application; Gamma function; Beta function; Bessel function; comprehensive

1. Introduction

Special function refers to some functions with specific properties, generally with established names and symbols, such as gamma function, beta function, Bessel function and so on. They play an important role in mathematical analysis, functional analysis, physical research and engineering application. Many special functions are solutions of differential equations or integrals of basic functions, so special functions often appear in the integral table, and integrals often appear in the definition of special functions. Traditionally, the analysis of special functions is mainly based on its numerical expansion. With the development of electronic computing, new research methods have emerged in this field.

Because special function is an important tool in mathematical analysis, the research and application of special function is very important. This paper summarizes the properties of special function, its application in integral operation and its application in physics, and draws some graphs of special function with Matlab software, mainly including defining properties, integral operation and its application in physics knowledge, and makes detailed exploration and proof with specific examples.

Definition and proof of special function

Learning special functions is a major difficulty and focus in mathematical analysis. Finding special functions contains many knowledge points and skills, and students can be guided to summarize through inquiry learning in teaching. On the one hand, it can improve students' skills and skills in finding function limits; On the other hand, it can also cultivate students' ability of observation, analysis and classification, which is very beneficial to students' study and thinking habits.

Learning the properties of special functions and their related calculations is often not solved by one method because of various types of questions, diverse methods and strong skills, and there is no fixed law to follow. It is the flexible use of various methods and the concentrated expression of various thinking methods, so it is difficult. The way to solve this problem mainly lies in mastering the characteristics of special functions and some basic methods. The related properties and applications of special functions are discussed with specific examples.

2. Properties and applications of gamma function

2. 1. 1 gamma function definition:

The general definition of gamma function is: this definition only applies to the region of, because this is the condition that the integral converges to t=0. Assuming that the domain of a function is an interval, two properties of the г function are discussed below.

2. 1.2 г function is continuous in the interval.

In fact, it is known that both false integral and infinite integral converge, so infinite integral converges uniformly in the interval. And the integrand function is continuous in the interval d, and the г function is continuous in the interval. So the г function is continuous at z point, because z is any point in the interval, so the г function is continuous in the interval.

2. 1.3, recurrence formula of gamma function

This relationship can be proved by the original definition of integral method, as follows:

This shows that when z is a positive integer n, it is factorial.

As can be seen from formula (4), it is a semimeromorphic function, and the singularities in a finite region are all first-order poles, and the poles are z=0,-1, -2, ..., -n, ....

2. 1.4 Integral with г function

2.2 the nature and application of beta function

2.2. Definition of1beta function:

This function is called the b function (beta function).

Given that a domain is a region, three attributes are discussed below:

Properties of beta function

2.2.2 Symmetry: =. Actually, there is.

2.2.3 Recursive formula:, in fact, by the partial integral formula,, there is.

that is

Through symmetry,

In particular, recursive formulas are applied continuously, including

that is

At that time, there was

This formula shows that the definitions of B function and г function are not formally related, but they are internally related. This formula can be extended to

2.2.4

The following simple formula is derived from the above formula:

2.2.5 integration with Beta function

Example 2.2. 1

Solution: Use

(Because it is an even function)

Example 2.2.2 Application of Beta Function in Multiple Integrals

Calculate, where is the closed area surrounded by these three straight lines,

Solution: Make a transformation, which maps the area into a square:. therefore

Using this function in the calculation process can easily solve the problem of finding the original function by general methods.

2.3 the nature and application of Bessel function

Definition of Bessel function

Bessel function: the second-order coefficient linear ordinary differential equation is called? The order of Bessel equation, where y is the unknown function of x,? Is an arbitrary real number.

2.3.2 Recursive formula of Bessel function

In Formulas (5) and (6), Formula 3 is obtained by elimination, and Formula 4 is obtained by elimination.

In particular, when n is an integer, it is obtained from equations (3) and (4):

By analogy, when n is a positive integer, it can be expressed by and.

because

By analogy, it can also be expressed by and. So when n is an integer, it can be expressed by sum.

2.3.3 Semi-odd Bessel functions are elementary functions.

Proof: We can know from the properties of г function.

According to recursive formula

Generally there is.

In which continuous actions of n operators are represented, for example

As can be seen from the above relationship, Bessel functions of semi-odd order (n is a positive integer) are all elementary functions.

2.3.4 The application of Bessel function in physics;

A new sampling theorem for fast convergence of spectral finite functions. According to the specific problem, the convergence speed can be adjusted by convolution method to achieve the expected effect, and the calculation will not be too complicated. It is an indispensable tool in communication technology to reconstruct the sampling theorem from the discrete sampling values of a function, which makes

It's called Fourier transform. Its inverse transformation is

If there is a positive number b, the spectrum of b is considered to be finite. For this kind of function, as long as the sampling interval, there are discrete sampling values (where z represents all integers: 0,) which can reconstruct the function.

This is Shannon sampling theorem. The generating function in Shannon sampling theorem is

Because the convergence speed of Shannon sampling theorem is not fast enough, if the Fourier transform of the maximum sampling interval characteristic function is allowed at this time:

The following sampling method introduces Bessel function into the sampling theorem, which is characterized by fast convergence and can be adjusted according to practical problems, so that the function can be determined more accurately from not too many sampled values.

Firstly, the sampling theorem is established.

Settings:

Where is the zero-order Bessel function. Constructor:

manufacture

After calculation:

Use partial integration and consider all Fourier transforms.

The convergence speed can be accelerated by the function convolution method, so that N can be appropriately selected according to specific problems to achieve the expected effect. This adjustable sampling theorem does not increase much computation. Take away:

analogously

After calculation:

After calculation:

Then there is: for example, Fourier transform,

Record by discrete sampling value

Therefore, it is self-evident that the convergence speed of sampling theorem is accelerated. In contrast, the amount of calculation does not increase, and n can control the convergence speed.

Example 2.4, using

Lemma: When?

while

Because it can't be expressed by elementary function, Newton-Leibniz formula can't be used to calculate the value of definite integral, so the following calculation formula is used.

Firstly, it is proved that the function satisfies the Dirichlet sufficient condition, and the Fourier series in the interval is expanded as follows:

( 1)

In ...

The power series expansion of the function is:

Then the expansion of power series is: (2)

From Lemma and (2)

(3)

Modified Bessel function by order

Where the function is a positive integer, take, and then (3) can become

(4)

By comparing the coefficients of (1)(4)

The integrand function is an even number function, so

The formula is proved.

3. Concluding remarks

This paper is about the research on the properties of special functions and their related calculations. By studying the properties of special functions and summarizing the related calculations, we can better master the application of special functions when we encounter related interdisciplinary subjects in our daily study, and we can apply different properties of special functions to prove and calculate for different examples, thus solving related problems more concisely and reasonably. The application of some special functions is not fixed and can be proved and calculated in many ways. When solving a problem, we should observe the structure and type of the problem and choose the simplest method to solve it.

References:

[1] Wang Zhuxi. Introduction to Special Functions [M]. Peking University Publishing House, 2000.5,90-91.

[2] Liu Yulian. Lecture on Mathematical Analysis (Volume II) [M]. Higher Education Press, 2003,331.

[3] Liu Yulian. Lecture on Mathematical Analysis (Volume II) [M]. Higher Education Press, 2003,331.

Kun Wang. Application of beta function in integral calculation. [J] Science and Technology Information, 20 12(34)

Wang Jilin. Special functions and mathematical and physical equations [M]. Shanghai Jiaotong University Press, 2000, 96-98.

Tao Tianfang. Fast sampling theorem expressed by special function [J]. Journal of Shanghai University (Natural Science Edition),1997,8 (4): 368-371.

Rao Congjun, Wang Cheng. Let mathematical modeling activities promote the reform of mathematics teaching [J]. Journal of Minzu University of China (Natural Science Edition), 2004,2.

Zhao Yibin. Solution of definite integral of a special function [J]. Journal of Institute of Disaster Prevention Technology, 20 10, 1(3):38-39.

[9] Lin Dong. Exploration of simplified formula? On the proof of a conjecture [J]. Teaching bulletin, 1992.2.

Li Dexin. Three methods for calculating definite integral by using symmetry principle [J]. Advanced Mathematics Research, 2004,7 (6): 41? 42.

[1 1] Zhai Zhongxin, Gong Dongshan. Teaching and learning of advanced mathematics [J]. Science Higher Education, 2004 (6): 29? 34.

[12] Hu function and application [J]. Journal of Harbin Normal University.2002,18 (4):12 ~15.

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