Some properties and applications of the gamma function in the field of mathematics

Yuze Chen

doi:10.1117/12.3036777

21 July 2024 Some properties and applications of the gamma function in the field of mathematics

Yuze Chen

Proceedings Volume 13219, Fourth International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2024); 132190Z (2024) https://doi.org/10.1117/12.3036777
Event: 4th International Conference on Applied Mathematics, Modelling and Intelligent Computing (CAMMIC 2024), 2024, Kaifeng, China

Abstract

In the 18th century, mathematicians were quite interested in the factorial of the non-natural numbers such as 1.5 and 2.5, because when we plot the factorials of the positive integers on to the coordinate system, we can find that it seems like it has a smooth curve that can path through it. So we can find a general formula of the function of the factorials, which is the gamma function. In the year of 1729, Euler found this function(Beta function) when he was at the age of 22. 𝛣(𝑥) = ∫^+∞ ₀ 𝑒^−𝑡 𝑡 ^𝑥𝑑𝑡 . To make it easier for the application, we added 1 to the independent variable and the new function is the gamma function. 𝛤(𝑥) = 𝛣(𝑥) = ∫^+∞ ₀ 𝑒^−𝑡 𝑡 ^𝑥𝑑𝑡 After it is found out by Euler, it has been studied by the mathematicians from several aspects. And it has been applied to the different field of mathematics. In this paper, we are going to discuss about the identities and the applications of the gamma function. I hope it could be useful for the researchers after. In the chapter 1, we will introduce the background and list out the properties of the Gamma function and the functions that is related to it, and we will proof some of the theorems. In the chapter 2, we will discuss about the usage of the Gamma function in the field of integrals. In the chapter 3, we will discuss about the applications of the Gamma function in statistics. In the chapter4, we will introduce some other applications in mathematics. In the chapter5, we will conclude and make some expectations.

(2024) Published by SPIE. Downloading of the abstract is permitted for personal use only.

Citation Download Citation

Yuze Chen "Some properties and applications of the gamma function in the field of mathematics", Proc. SPIE 13219, Fourth International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2024), 132190Z (21 July 2024); https://doi.org/10.1117/12.3036777

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KEYWORDS

Mathematics

Statistical analysis

Integration

Optical spheres

Probability theory

Lead

Matrices

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