An algebraic formulation of the simplex linear method for multiple integrals over the d-dimensional domain

We present the systematic treatment of multiple integrals over a simplex of the dimensionality d (d-simplex). The integrand is an analytical function in a suitable domain, and the linear form was employed as an argument. Using the elements obtained from the values of the integrand at the vertices of the d-simplex, integral can be evaluated analytically and expressed as the quotient of two determinants. Potential applications of the results described in the present study include numerical computations of Greenʼs function (GF), and the response of quantum systems in terms of complex susceptibilities in multidimensional space.

► Integral over the d-dimensional simplex.
► Analytical properties of the integrand function on the complex plane.
► The integral solution in the form of the quotient of two determinants.