Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics
Pranesh Kulkarni
Dr. Pranesh Kulkarni, Assistant Professor, Department of Mathematics, T. John Institute of Technology, Bannerghatta Road Gottigere Bangalore (Karnataka), India.
Manuscript received on 10 March 2025 | First Revised Manuscript received on 21 March 2025 | Second Revised Manuscript received on 05 April 2025 | Manuscript Accepted on 15 April 2025 | Manuscript published on 30 April 2025 | PP: 35-37 | Volume-5 Issue-1, April 2025 | Retrieval Number: 100.1/ijam.A119505010425 | DOI: 10.54105/ijam.A1195.05010425
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Abstract: In many areas of applied mathematics, various types of Special functions have become essential tools for Scientists and engineers. Both Beta and Gamma functions are crucial to calculus, as complex integrals can be reduced to a simpler form. In physics and engineering, problems require a detailed knowledge of applied mathematics and an understanding of special functions, such as the gamma and beta functions. The topic of special functions is critical and constantly expanding with the emergence of new problems in the applied sciences. In this article, we describe the basic theory of gamma and beta functions, their interconnections, and their applications to engineering problems.to compute and depict scattering amplitude in Reggae trajectories. We aim to demonstrate that the extension of the classical beta function has numerous applications. It helps in providing new extensions of the beta distribution, as well as new extensions of the Gauss hypergeometric functions and the confluent hypergeometric function, generating relations, and extending Riemann-Liouville derivatives. In this Article, we develop some elementary properties of the beta and gamma functions. We provide multiple proofs for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogy of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We demonstrate how they can be used to prove Fermat’s theorem, which states that a prime of the form 4n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral,
Keywords: Beta Function, Gamma Function, A Pochhammer symbol in Applied Mathematics, Engineering.
Scope of the Article: Applied Mathematics