Extension of Hölder’s theorem

Abstract

This report attempts to explore and extend the use of Otto Hölder’s theorem on the Gamma Function, Γ, and apply it to different cases. In doing so, we will see how the theorem holds true in these cases and it will be easier and more convenient to apply the Gamma Function in different situations. The Gamma Function and Hölder’s theorem will be explained and elaborated on and the author goes on to explain the proof of the theorem. The author then goes on to analyse the cases of Γ(z+1) = zΓ(z), Γ(zn+1) = znΓ(zn) and Γ(-z) in the Gamma Function. The author also explores extending Hölder’s theorem to the Riemann Zeta Function and the Beta Function that incorporates the Gamma Function in their application after elaborating and explaining further these two functions. The application of these functions as well as their background and history will also be provided to have a deeper understanding of the importance of the Gamma Function and Hölder’s theorem in mathematical development. It is clear that a deeper understanding and appreciation of the use of these theorems, formulas and functions will provide for not only a more convenient use in their application but the author hopes that through this report, the reader will be intrigued to go on and further study and develop the functions and theorems that are discussed in this report. Further study of different cases of the Gamma Function should be anticipated as the function is incorporated into many probability distributions and is widely used for statistical analysis. It is important to continue to develop and analyse how Hölder’s theorem holds true in the different functions that incorporate the Gamma Function to make using these functions more convenient in the future.