Study of Hypergeometric Functions in Context of Nonelementary Integrals

Abstract

The present paper is a study of relations between elementary functions, nonelementary functions, nonelementary integrals and hypergeometric functions in context of antiderivatives. Antiderivative is one of the important mathematical operator or machine, which produces new functions. That new function may or may not belong to the set from which the integrand might have been taken. The problem arises, when the new function doesn’t belong to the previous set, the set of elementary functions, which opens the scope for the development of new functions. Till the new function is propounded, such integral, which doesn’t belong to the previous set of elementary functions, are treated as nonelementary function or nonelementary integral. The introduction of special functions like hypergeometric function has ended the study of searching new elementary and nonelementary functions, which has great effect on expressing antiderivatives of those integrands, whose integrals are still not expressible in terms of elementary functions. Due to the less research on it, has almost terminated the development of new properties between such functions in context of elementary and nonelementary functions. In this paper an attempt has been made to propound some propositions between elementary and nonelementary functions with special functions based on sufficient examples in context of nonelementary integrals. The whole discussion is centred about five questions: are all elementary functions hypergeometric functions?, are all nonelementary functions hypergeometric functions?, are all hypergeometric functions elementary functions?, are all hypergeometric functions nonelementary functions?, and are all hypergeometric functions as output from Mathematica as antiderivative nonelementary functions? Studying on these queries we propounded eleven propositions, out of which one of the important finding is that the outcome of the input “Integrate[f[x], x]” from Mathematica in terms of hypergeometric functions is always nonelementary functions (or integrals) and every elementary and nonelementary functions cannot be expressed in terms of hypergeometric functions. We also found that every real (or complex) numbers can be expressed in terms of hypergeometric function. The paper ends with answers of the five questions and also opens a new scope of research for the computer software experts to explore new ideas of elementary and nonelementary functions in terms of antiderivatives.