## A Study Of Indefinite Nonintegrable Functions

**Abstract Category:**

**Science**

**Course / Degree:**Ph. D.

**Institution / University:**Vinoba Bhave University, Hazaribag, Jharkhand, India, India

**Published in:**2012

**Thesis Abstract / Summary:**

The present thesis is intended to categories many nonintegrable functions into some standard forms and then to find their indefinite integrals in terms of dominating functions. We have divided the thesis into eight chapters followed by its conclusion and the scope of future work in ninth chapter, whose abstracts are as follows:

Chapter-I Introduction

The chapter starts with the definition of elementary function and its indefinite integration followed by the range and difficulty of the problem of indefinite integration. We have discussed some functions beyond the region of elementary functions, which have already been proved nonelementary (indefinite nonintegrable) by the pioneers of the subject.

Some previous search of algorithms for elementary and nonelementary functions (indefinite integrals) by Bernoulli, Laplace, Abel, Liouville, Marchisotto & Zakeri, etc. has been discussed. We have also mentioned some new functions originated from the indefinite nonintegrable functions, some drawbacks present in the previous works, the objective of the work, and the methodology applied in the thesis. The chapter ends with a short note on the significance of the research work.

Chapter-II Six Conjectures on Indefinite Nonintegrable Functions

In this chapter we have introduced six standard forms of indefinite nonintegrable functions as six conjectures with some examples. We proved them indefinite nonintegrable by the help of strong Liouville theorem, the special case of strong Liouville’s theorem, and some properties due to Marchisotto & Zakeri.

Chapter-III Dominating Function

In the chapter we have introduced a dominating function, which dominates all most all the functions like algebraic, exponential, trigonometric, hyperbolic, logarithmic, etc. Then a list of well known Taylor’s series or Laurent’s series expansions of different elementary functions about the point x=0 is also provided to prove that all these functions are d-able functions. We have also discussed the convergence, interval and radius of convergence of a dominating function. The chapter ends with a property named as continuity, differentiability and integrability of a dominating function.

Chapter-IV Types of Dominating Functions

Applying the concept of d-function, we have extended the class of elementary functions by introducing some more dominating functions related to trigonometric, hyperbolic, exponential and logarithmic functions, which dominates the classical elementary trigonometric, hyperbolic, exponential and logarithmic functions in the sense that for particular values of ‘m’ and ‘r’, the introduced d-functions reduce to the classical elementary functions.

Chapter-V Dominating Sequential Functions

In this chapter we have introduced some new functions originated from the classical trigonometric, hyperbolic, exponential, logarithmic and all dominating functions by taking their inner product with some sequences. The functions originated from the classical functions have been called sequential functions and those originated from the dominating functions have been called dominating sequential functions.

Chapter-VI Indefinite Integrals of Dominating Sequential Functions

In this chapter we have generated the general integrals of dominating sequential functions and then deduced the indefinite integrals of classical trigonometric, hyperbolic, exponential, and logarithmic functions as well as of some classical nonintegrable functions.

Chapter-VII Existence Theorems on Indefinite Integrability

In this chapter we have propounded a necessary condition, a sufficient condition as well as a necessary and sufficient condition for the existence of indefinite integral of a function on the basis of d-function theory. Thereafter we compared the propounded conditions with the classical one and presented a modified sufficient condition for it.

Chapter-VIII Possible Integrals of Indefinite Nonintegrable Functions

The aim of this chapter is to give possible integrals of the indefinite nonintegrable functions discussed in chapter two. To find the integrals we have first checked their continuity or piecewise continuity by their graphs using software ‘Graph Version 4.3 Build 384’, to satisfy the classical sufficient condition for integrability.

Thereafter we have found their power series using Taylor’s series about the point x=0 by the help of the online software ‘Wolfram Alpha Computational Knowledge Engine’ in the form of d-function to prove that they are d-able functions, so that we can integrate them term by term.

We have tried to denote the integrals in the form of different dominating, sequential and dominating sequential functions. The integrals not coming under any of the dominating functions form have been left in series.

Chapter-IX Conclusion and Future Work

In this chapter we have concluded the thesis work and have mentioned the scope of the future work.

We have also included several references in the hope that the readers will take help from these for future work followed by a list of publications and a list of research papers presented in the conferences related to this work.

The thesis ends with short notes on extension of partial fractions and term by term integration in appendixes I and II respectively.

**Thesis Keywords/Search Tags:**

Nonelementary Functions, Conjectures, Dominating Sequential Functions, Dominating Function, Sequential Functions

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**Submission Details**: Thesis Abstract submitted by **Dharmendra Kumar Yadav** from **India** on 23-Sep-2016 17:30.

Abstract has been viewed 795 times (since 7 Mar 2010).

**Dharmendra Kumar Yadav Contact Details:** Email: mdrdkyadav@gmail.com Phone: +919891643856

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