Applications of Neutrix to the beta function

The objective of this thesis is to extend the domains of the beta function and its partial derivatives for all real numbers to get some results on these functions. For these purpose, neutrix calculus developed by Van der Carput, have been used. The thesis consist of four chapters. In the second section after introduction, gamma function, beta function and its partial derivatives are given with some properties. Also, the concepts of neutrix and neutrix limit are given with examples. In the third chapter, the definition of the beta function $B(\eta,\theta)$ and its partial derivatives have been given for all real values of $\eta$ and $\theta$, by using the concept of neutrix and neutrix limit. In the last section, some results are given for the beta function $B(\eta,\theta)$ and its partial derivatives for negative integer values of $\eta$ and $\theta$.