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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 12H: Differential and difference algebra

Introduction

History

Applications and related fields

For applications of Differential fields see 34: Ordinary Differential Equations and 35: Partial Differential Equations.

General-purpose symbolic algebra programs (see 68D30) tacitly perform computations in fields, including Z/pZ. The fairly sophisticated products also carry out algorithms from differential algebra to determine the integrability of elementary functions.

Subfields

• 12H05: Differential algebra [See also 13Nxx]
• 12H10: Difference algebra [See also 39Axx]
• 12H20: Abstract differential equations [See also 34Mxx]
• 12H25: p-adic differential equations [See also 11S80, 14G20]
• 12H99: None of the above but in this section

Parent field: 12: Field theory and polynomials

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

• Magid, Andy R. "Lectures on differential Galois theory", University Lecture Series, 7. American Mathematical Society, Providence, RI, 1994. 105 pp. ISBN 0-8218-7004-1 MR95j:12008
• Mikhalëv, A. V.; Pankrat´ev, E. V., "Differential and difference algebra" (in Russian) Algebra. Topology. Geometry, Vol. 25 (Russian), 67--139, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987. MR89a:12011

Software and tables

Handbooks of integrals are common; particularly massive is the set of integral tables by Gradshteyn, I.S. and Ryzhik, I.M. "Tables of Integrals, Series, and Products", (5th ed, 1993), San Diego CA: Academic Press. Somewhat closer to a textbook (offering some discussion of the principal themes) is Zwillinger, Daniel: "Handbook of integration", Jones and Bartlett Publishers, Boston, MA, 1992. ISBN 0-86720-293-9.

Online integrators from Wolfram Inc. and Fateman. (The latter calls the former if it gets stuck.)

Symbolic Summation package for Mathematica

Other web sites with this focus

Selected topics at this site

• Closed form integration: the case of algebraic integrands
• Reducing some non-elementary algebraic antiderivatives to elliptic integrals (Riemann surfaces)
• Rewriting trigonometric integrals as algebraic (elliptic) ones
• References and pointers to the Risch algorithm and symbolic integration
• Is there a closed-form "solution" for an elliptic integral? (no)
• What functions have antiderivatives which are elementary functions ? Citations and long article by Matthew Wiener. (Includes topics in symbolic integration.) Frequently-mentioned integrands with no elementary antiderivative include exp(-x^2), sin(x)/x, x^x, sqrt(1-x^4), and many variants.
• Proving some functions have no elementary antiderivative [update of previous article]
• Comparisons of deterministic and heuristic algorithms to decide whether a function has an elementary antiderivative (Risch algorithm, popular software, etc.)
• Maple incorporates algorithms to integrate functions from a genus-0 extension of the rational function field
• Why isn't the Risch algorithm fully implemented in computer algebra systems?
• Humans can integrate 1/(1+tan(x)^c) on [0,pi/2] but machines cannot!
• Formal characterization of "closed-form solutions" to differential equations, integrals, and summation problems
• Typical symbolic integration showing the need to assist faulty software (Maple)
• Critique of domain restrictions in display of integration results (Mathematica)