A computational method for hypersurfaces in terms of gamma function
The author provides a general volume formula for regions often considered in physical and engineering problems. Dirichlet integrals in terms of beta and gamma functions can be obtained by performing a suitable variable transformation from Cartesian to polar coordinates and vice versa. The main idea is to extend the computation of such integrals having 2D integration domain to find volumes of a wide range of \(n\)-dimensional axially symmetric regions given by the equation \[ G(x)=\sum_{i=1}^n\biggl|{{x_i}\over{a_i}}\biggr|^{\alpha_i}-1=0, \] \(x_i\) real, \(\alpha_i\) and \(a_i,\;i=1,\ldots,n\) positive constants. First volume of regions bounded by hypersurfaces are considered. Then a generalization to more elaborate shapes is presented. Finally an algorithm for the numerical computation of such volumes is proposed, where an asymptotic expansion of the gamma function is given by Stirling's formula.
scientific article; zbMATH DE number 2136878