1, product property: the product property of gamma function can be expressed as gamma (a), gamma (b)= gamma (a+b). This property is very useful in solving some mathematical problems, because it allows us to simplify the multiplication result of two gamma functions into one gamma function.
2. Reflective property: The reflective property of γ function can be expressed as the power of γ(x)γ( 1-x)=π sin(πx). This property is very important in dealing with some mathematical problems involving gamma function, because it allows us to find another part through one known part.
3. Integral property: the gamma function is continuous, that is, for any real number x and positive integer n, there is gamma (x+n)= gamma (x)*n! . This property enables us to simplify a complex function integration problem into a series of simple function integration problems.
Main applications of gamma function:
1, Combinatorial Mathematics: Gamma function plays an important role in Combinatorial Mathematics. Combinatorial mathematics is a subject that studies discrete structures and corresponding mathematical problems, such as permutation, combination, division and division. Gamma function can be used to solve some combinatorial problems, such as calculating the sum of factorials and solving combinatorial identities.
2. Probability statistics: Gamma function is also widely used in probability statistics. It can be used to describe the distribution of continuous random variables such as exponential distribution, gamma distribution and beta distribution. Gamma function can also be used to solve some probability problems, such as calculating the probability of events and solving the digital characteristics of random variables.
3. Special functions and integrals: Gamma functions can be used to solve some special functions and integrals. For example, using the definition and properties of gamma function, we can solve some difficult integration problems, such as solving Gaussian integral and Bessel function. In addition, the gamma function can also be used to solve the value of some special functions, such as solving the value of factorial and solving the value of Gaussian polynomial.