Some results on the incomplete beta function

Objective: This thesis aims to define the beta and incomplete beta functions, as well as the partial derivatives of these functions, over a broader domain by using neutrix calculus. Additionally, the theorems and results derived from the definitions of the partial derivatives of the beta and incomplete beta functions will be presented. Material and Methods: All articles and books related to the topic constitute the research materials. As a research technique, the articles listed in the references section were reviewed, and thereafter the study was conducted using mathematical proof methods. Results: This thesis consists of four chapters. Following the introduction, the second chapter provides summaries of the sources used in the thesis. In this chapter, the concepts of the beta function, incomplete beta function, neutrix, and neutrix limit are introduced. Then, the definitions of the partial derivatives of the beta and incomplete beta functions are given, and how these functions and their partial derivatives can be defined for negative integer values is discussed. In the third chapter, theorems and results obtained for certain values of the beta and incomplete beta functions and their partial derivatives using the concept of neutrix are presented. Conclusion: In this thesis study, the beta function and the incomplete beta function are introduced, and their properties are examined in detail with the aid of neutrix calculus. Additionally, the theorems and results obtained through these definitions are presented.