THE GAMMA AND BETA MATRIX FUNCTION AND OTHER APPLICATIONS

Abstract

The paper presents Gamma and Beta functions and their
applications to real life problems. After relating the gamma function with
Euler’s infinite products, the Hermite series and Hypergeometric series, then the computation of perimeter ( n p ) of a polygon involving gamma function by which a factor 2S n p exceeds the perimeter of a unit circle is presented in the sense of Jorda and Cortise. Application to Hermite-Laguerre polynomial and multivariate calculus Hypergeometric matrix functions are presented with convergence of gamma matrix using the Ratio Test. The procedure for detecting nearness to singularity of the gamma matrix is described in terms of condition number where eigenvalues are ordered according to their magnitudes. In addition, the Numerical radius of the Gamma matrix is introduced which helps in the computation of a bound for the condition number of the matrix. In particular, we paid special attention to the analysis of the pendulum problem as a second order differential initial value problem wherein, Jacobi elliptic integrals of first and second kinds play major roles. The bounds for these elliptic integrals are discussed in details using some ideas in the existing literatures. It is established in this paper that, there exists no universally most acceptable bound for these Jacobi elliptic integrals as attested to by various authors. It is therefore suggested in this paper that these bounds may be subjected to probabilistic analysis in our future work. It is also hoped to link these bounds for the Jacobi elliptic integrals with the Weierstrass elliptic
functions as well.