Integration in Finite Terms: Fundamental Sources

herausgegeben von: Dr. Clemens G. Raab, Dr. Michael F. Singer

Verlag: Springer International Publishing

Buchreihe : Texts & Monographs in Symbolic Computation

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Inhaltsverzeichnis

Über dieses Buch

This volume gives an up-to-date review of the subject Integration in Finite Terms. The book collects four significant texts together with an extensive bibliography and commentaries discussing these works and their impact. These texts, either out of print or never published before, are fundamental to the subject of the book. Applications in combinatorics and physics have aroused a renewed interest in this well-developed area devoted to finding solutions of differential equations and, in particular, antiderivatives, expressible in terms of classes of elementary and special functions.

Inhaltsverzeichnis

Frontmatter

Integration in Finite Terms

Abstract

The Question arises in elementary calculus: Can the indefinite integral of an explicitly given function of one variable always be expressed "explicitly" (or "in closed form", or "in finite terms")? Liouville gave the answer one would expect, "No", and he proved in particular that such is not the case with \(\int {e}^{{x}^{2}}\) dx. Since we have all fallen into the habit of quoting this result and giving neither proof nor reference, it may be worthwhile to actually state it as precisely as possible and give a proof that is as elementary as the subject matter might suggest.

Maxwell Rosenlicht

Comments on Rosenlicht’s Integration in Finite Terms

Abstract

In 1968, Maxwell Rosenlicht [Ros68] published the first purely algebraic proof of Liouville’s Theorem on Integration in Finite Terms (which we will simply refer to as “Liouville’s Theorem”) . This paper, together with Robert Risch’s paper [Ris69], stimulated renewed interest in both the mathematical and algorithmic aspects of this area. The paper Integration in Finite Terms [Ros72] appearing in this volume presents the material of [Ros68] in a simplified form, suitable for an advanced undergraduate.

Michael F. Singer

Integration in Finite Terms Liouville’s Theory of Elementary Methods

Abstract

The functions which we shall study in the present chapter are essentially those which make up the functional world of a student of the integral calculus. Such a student, if not familiar with the concept of algebraic function in its most general form, knows the polynomials and fractional rational functions, has seen functions involving radicals, and can imagine quite well the most general algebraic function which can be expressed in terms of radicals. He knows e^x, log x, sin x, cos x, and the inverses of the latter two functions.

Joseph Fels Ritt

Comments on J.F. Ritt’s Book Integration in Finite Terms

Abstract

I saw J.F. Ritt’s book [Rit48] for the first time in 1969 when I was an undergraduate student. I had just started to work on topological obstructions to the representability of algebraic functions by radicals and on an algebraic version of Hilbert’s 13th problem on the representability of algebraic functions of several complex variables by composition of algebraic functions of fewer number of variables. My beloved supervisor Vladimir Igorevich Arnold was very interested in these questions.

Askold Khovanskii

On the Integration of Elementary Functions which are Built Up Using Algebraic Operations

Abstract

This paper gives an algorithm to decide if a function built up from the rational functions using logarithms, exponentials and arbitrary algebraic operations can be integrated in terms of functions built up in a similar manner.

Robert H. Risch

Comments on Risch’s On the Integration of Elementary Functions which are Built Up Using Algebraic Operations

Abstract

In the two reports [Ris68, Ris69a], Risch devised a recursive algorithm that decides when a given elementary function has an elementary integral. At that time, only the complete algorithm for transcendental elementary integrands was published [Ris69b]. By abuse of language, “transcendental” elementary integrands is a short way of referring to the more restrictive class of elementary integrands that lie in differential fields whose generators are algebraically independent over the constants.

Clemens G. Raab

Integration of Algebraic Functions

Abstract

This thesis will provide a practical decision procedure for the indefinite integration of algebraic functions. Unless explicitly noted, we will always assume x to be our distinguished variable of integration. An algebraic function y of x is defined as a solution of a monic polynomial in y with coefficients that are rational functions in x. Each of these rational functions in x can be written as a quotient of polynomials in x whose coefficients are constants (i.e. not dependent on x).

Barry M. Trager

Comments on Integration of Algebraic Functions

Abstract

We survey some more recent work on some of results presented in the thesis on Integration of Algebraic Functions [33]. The thesis was intended to provide practical algorithms for performing the integration of algebraic functions after the fundamental theoretical results of Risch in [29] and [30] proving that the problem was solvable. At roughly the same time, James Davenport developed and implemented his approach to the problem [16, 15]. Following the local approach suggested by Risch, Davenport developed implementations of Puiseux expansions and Coates’ algorithm to construct multiples of a divisor [11].

Barry M. Trager

Titel: Integration in Finite Terms: Fundamental Sources
herausgegeben von: Dr. Clemens G. Raab
Dr. Michael F. Singer
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-98767-1
Print ISBN: 978-3-030-98766-4
DOI: https://doi.org/10.1007/978-3-030-98767-1