Analytical expression for the exact curved surface area of a frustum of hemiellipsoid, through hypergeometric function approach

TL;DR

This paper derives a new analytical formula for the exact curved surface area of a frustum of hemiellipsoid using hypergeometric functions, expanding mathematical tools for geometric surface calculations.

Contribution

It introduces a novel closed-form expression involving Appell's function and triple hypergeometric series, not previously documented in literature.

Findings

Derived an explicit formula for the surface area

Validated the formula numerically with Mathematica

Enhanced mathematical methods for geometric surface analysis

Abstract

Our present investigation is motivated essentially by several interesting applications of generalized hypergeometric functions of one, two and more variables. The hypergeometric functions are potentially useful and have widespread applications related to the problems in the mathematical, physical, engineering and statistical sciences. In this article, we aim at obtaining the analytical expression (not previously found and recorded in the literature) for the exact curved surface area of a frustum of hemiellipsoid in terms of Appell's function of second kind and general triple hypergeometric series of Srivastava. The derivation is based on Mellin-Barnes type contour integral representations of generalized hypergeometric function$~_pF_q(z)$, Meijer's $G$-function and series manipulation technique. The closed form for the exact curved surface area of a frustum of hemiellipsoid is also verified numerically by using {\it Mathematica Program}.