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Journal ArticleDOI

The solution of the problem of integration in finite terms

01 May 1970-Bulletin of the American Mathematical Society (American Mathematical Society)-Vol. 76, Iss: 3, pp 605-608
TL;DR: In this paper, an algorithm for deciding whether an elementary function has an elementary indefinite integral and for finding the integral if it does is presented. But the problem of integration in finite terms has several precise, but inequivalent formulations.
Abstract: Introduction. The problem of integration in finite terms asks for an algorithm for deciding whether an elementary function has an elementary indefinite integral and for finding the integral if it does. "Elementary" is used here to denote those functions built up from the rational functions using only exponentiation, logarithms, trigonometric, inverse trigonometric and algebraic operations. This vaguely worded question has several precise, but inequivalent formulations. The writer has devised an algorithm which solves the classical problem of Liouville. A complete account is planned for a future publication. The present note is intended to indicate some of the ideas and techniques involved.

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Citations
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BookDOI
01 Jan 2003
TL;DR: In this paper, a large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, inverse problem and linear differential equations in positive characteristic.
Abstract: Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

942 citations


Cites background from "The solution of the problem of inte..."

  • ...In 1970, Risch [245] showed that this problem could be solved if one could decide if a given divisor on a given algebraic curve is of finite order....

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Journal ArticleDOI
TL;DR: This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems.
Abstract: Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

937 citations


Cites background or methods from "The solution of the problem of inte..."

  • ...For rational functions with only simple poles, this algorithm first appears in Johann Bernoulli (1703). For symbolic computation, this approach is inefficient since it involves polynomial factorization and computation with algebraic numbers, and most computer algebra systems implement the algorithms described in this chapter. In fact, on the pages just preceding Bernoulli’s article, Leibniz (1703) goes a step further....

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  • ...For information about the highly active area of symbolic-numeric computations see the Special Issue of the Journal of Symbolic Computation (Watt & Stetter 1998) and Corless, Kaltofen & Watt (2003). 2.4. The first algorithm for division with remainder of polynomials appears in Nuñez (1567). He is, of course, limited by the concepts of his times to specific degrees, 3 and 1 in his case, and positive coefficients....

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  • ...Knuth (1993) explains Johann Faulhaber’s (1631) methods for summation of powers, yielding much more beautiful expressions than the ones we give....

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  • ...π had been used by Jones (1706) and by Christian Goldbach in 1742....

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  • ...Knuth (1993) explains Johann Faulhaber’s (1631) methods for summation of powers, yielding much more beautiful expressions than the ones we give. A true renaissance man. 23.3. The definition of the gff, as well as Theorem 23.12 and Algorithm 23.13, are from Paule (1995). We have adopted the graphical representation of the shift structure from Pirastu (1992)....

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Journal ArticleDOI
TL;DR: In this paper, it is shown that it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals, which allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.
Abstract: It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Sometimes, though, it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals. These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.

279 citations

Book ChapterDOI
TL;DR: In this article, the authors present differential algebraic techniques for differential algebraic geometry and differential algebraic techniques are used to calculate properties of fields and Spectrometers.
Abstract: Dynamics. Differential Algebraic Techniques. Fields. Maps: Calculation. Maps: Properties. Spectrometers. Repetitive Systems.

261 citations

Journal ArticleDOI
Michael Karr1
TL;DR: Results which allow either the computatmn of symbolic solutions to first-order-linear difference equauons or the determination that solutions of a certain form do not exist are presented.
Abstract: Results which allow either the computatmn of symbolic solutions to first-order-linear difference equauons or the determination that solutions of a certain form do not exist are presented. Starting with a field of constants, larger fields may be constructed by the formal adjunctlon of symbols whtch behave hke solutions to first-order-linear equations (with a few restrictions) It IS in these extension fields that the difference equations may be posed and m which the solutions are requested. The principal apphcatmn of these results is In finding formulas for a broad class of finite sums or In showing the nonexistence of such formulas.

224 citations

References
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Journal ArticleDOI
TL;DR: In this paper, the authors give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. They also give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logariths can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied.
Abstract: This paper deals with the problem of telling whether a given elementary function, in the sense of analysis, has an elementary indefinite integral. In ?1 of this work, we give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. By using functions of a complex, rather than a real variable, we can limit ourselves to exponentiation, taking logs, and algebraic operations in defining the elementary functions, since sin, tan- 1, etc., can be expressed in terms of these three. Following Ostrowski [9], we use the concept of a differential field. We strengthen the classical Liouville theorem and derive a number of consequences. ?2 uses the terminology of mathematical logic to discuss formulations of the problem of integration in finite terms. ?3 (the major part of this paper) uses the previously developed theory to give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logarithms can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied. The man who established integration in finite terms as a mathematical discipline was Joseph Liouville (1809-1882), whose work on this subject appeared in the years 1833-1841. The Russian mathematician D. D. Mordoukhay-Boltovskoy (1876-1952) wrote much on this and related matters. The present writer received his introduction to this subject through the book [10] by the American J. F. Ritt

321 citations

Journal ArticleDOI

237 citations


"The solution of the problem of inte..." refers background in this paper

  • ...(1), (2) and (3) are all the operations needed since we get the trigonometric and inverse trigonometric operations by adjoining V ( - l ) toK....

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  • ...If in (2) or (3) 0t- is transcendental over 3D(0i, • • • , 0,_i) and the constant field of 3D(0i, • • • ,0;_i) is the same as that of 3D(0i, • • • , 0*), then 0» is a monomial over 3D(0i, • • • , 0t-x)....

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  • ...If 3) is a differential subfield of $, then ^ (and any /G*?) is said to be elementary over 3D iff 3F = 3D(0i, • • • , 0n) where each 0t- satisfies a t least one of the following conditions: (1) 6i is algebraic over 3D(0i, • • • , 0,~i), (2) 9'i/di•=ƒ' for some/G3D(0x, • • • , 0i_i) (the exponential case), (3) f/f=0't for some ƒ G 3D (0i, • • • , 0*_i) (the logarithmic case)....

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Book
01 Jan 1933

162 citations