Analytical expression for the exact curved surface area and volume of a hyperboloid of one sheet via Mellin-Barnes type contour integration

TL;DR

This paper derives new exact analytical formulas for the surface area and volume of a hyperboloid of one sheet using Mellin-Barnes contour integrals and hypergeometric functions, verified numerically.

Contribution

It provides the first explicit formulas for hyperboloid surface area and volume in terms of hypergeometric functions, expanding mathematical tools for geometric analysis.

Findings

Exact formulas for surface area and volume derived

Formulas verified numerically with Mathematica

Uses Mellin-Barnes contour integral and hypergeometric functions

Abstract

In this article, we aim at obtaining the analytical expression ({\bf not previously found and recorded in the literature}) for the exact curved surface area of a hyperboloid of one sheet in terms of Srivastava-Daoust triple hypergeometric function. The derivation is based on Mellin-Barnes type contour integral representations of generalized hypergeometric function$~_pF_q(z)$, Meijer's $G$-function, decomposition formula for Meijer's $G$-function and series rearrangement technique. Further, we also obtain the formula for the volume of a hyperboloid of one sheet. The closed forms for the exact curved surface area and volume of the hyperboloid of one sheet are also verified numerically by using {\it Mathematica Program}.