Some Approximation Results And It's Applications About Gamma Function And Related Functions

As an important research problem,approximation of Gamma function and it's related func-tions have been researched for a long time and have been widely used in many fields.For example,for some complex integral representation forms in probability theory,as well as math-ematical problems such as function distribution and digital characteristics,the study of Gamma approximation function can solve them effectively and conveniently.The relationship between Gamma function and some special distribution functions can be used to construct the asymptot-ic expansion of distribution density functions.The approximation problem of Gamma function ratio can also be applied to the study of Mellin-Barnes type integrals and hypergeometric func-tions.In addition,we can get the approximate distribution of test statistics in various hypothesis testing problems by using the product form of gamma function ratio.So it is very important to study the Gamma function and it's related functions.In this dissertation,we study Gamma function approximation,Digamma function approx-imation and the ratio of Gamma function approximation.Our results have fast convergence speed,accurate approximation and can be applied to some practical problems.In chapter 2,based on Burnside formula and Ramanujan formula,we construct two main forms of Gamma function approximation and improve them by using continued fraction type remainder approx-imation method to improve the accuracy of Gamma function approximation.Meanwhile,we give the corresponding two-side inequalities.Then,by using the relationship between Stirling series approximation formula and F distribution density function,we construct the asymptotic expansion form of F distribution density function with freedom degree(8),9))(9)??).In chapter 3,we construct a relation between Digamma function and Trigamma function by us-ing Lagrange's median theorem.Then we also get the approximation result about the ratio of Gamma function.The approximation of Gamma function ratio is further applied to the research of ₂₁((6,(7,(8;)when(8??.Finally,we give some numerical computations to compare the approximation of our results with the known approximation formulas.We can see that our results have faster convergence rate and are more superior.