Analytical expressions for the exact curved surface area of a hemiellipsoid via Mellin-Barnes type contour integration

TL;DR

This paper derives new analytical formulas for the exact curved surface area of a hemiellipsoid using Mellin-Barnes contour integration and hypergeometric functions, including special cases for related shapes.

Contribution

It provides the first explicit analytical expressions for the surface area of a hemiellipsoid in terms of Appell's hypergeometric functions, not previously recorded in literature.

Findings

Derived new closed-form expressions for hemiellipsoid surface area.

Validated formulas numerically using Mathematica.

Extended results to special cases like ellipsoid and spheroids.

Abstract

In this article, we aim at obtaining the analytical expressions ({\bf not previously found and not recorded in the literature}) for the exact curved surface area of a hemiellpsoid in terms of Appell's double hypergeometric function of first kind. The derivation is based on Mellin-Barnes type contour integral representations of generalized hypergeometric function$~_pF_q(z)$, Meijer's $G$-function and analytic continuation formula for Gauss function. Moreover, we obtain some special cases related to ellipsoid, Prolate spheroid and Oblate spheroid. The closed forms for the exact curved surface area of a hemiellpsoid are also verified numerically by using {\it Mathematica Program}.