Addition theorem of Abel type for hyper-logarithms

Abstract

Several kinds of generalizations of classical Abel Theorem in algebraic curves are known, for example see [12] and [13]. It seems to the author these are all regarded as local relations among rational differential forms. In this article we shall try to generalize Abel Theorem for integrals of rational forms in some specific cases where these can be described in terms of hyper-logarithms (for the definition see [3] and [4], Theorem 2). Trigonometric functions have been generalized to higher dimensional cases by L. Schläfli who has obtained a very important variational formula related to the volume of a spherical simplex [16].