The Risch differential equation problem
SIAM Journal on Computing
Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
Journal of Symbolic Computation
Symbolic integration: the stormy decade
Communications of the ACM
Formal solutions of differential equations
Journal of Symbolic Computation
Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
Journal of Symbolic Computation
An algorithms for computing integral bases of an algebraic function field
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
The Risch differential equation on an algebraic curve
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
On solutions of linear ordinary differential equations in their coefficient field
Journal of Symbolic Computation
Computer algebra handbook
Algebraic general solutions of algebraic ordinary differential equations
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Solving second order linear differential equations with Klein's theorem
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Computing monodromy groups defined by plane algebraic curves
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Symbolic summation with radical expressions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Integration of algebraic functions: a simple heuristic for finding the logarithmic part
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Effective Set Membership in Computer Algebra and Beyond
Proceedings of the 9th AISC international conference, the 15th Calculemas symposium, and the 7th international MKM conference on Intelligent Computer Mathematics
Good reduction of Puiseux series and applications
Journal of Symbolic Computation
| Hi-index | 0.00 |
We extend a recent algorithm of Trager to a decision procedure for the indefinite integration of elementary functions. We can express the integral as an elementary function or prove that it is not elementary. We show that if the problem of integration in finite terms is solvable on a given elementary function field k, then it is solvable in any algebraic extension of k(@d), where @d is a logarithm or exponential of an element of k. Our proof considers an element of such an extension field to be an algebraic function of one variable over k. In his algorithm for the integration of algebraic functions, Trager describes a Hermitetypereduction to reduce the problem to an integrand with only simple finite poles on the associated Riemann surface. We generalize that technique to curves over liouvillian ground fields, and use it to simplify our integrands. Once the multiple finite poles have been removed, we use the Puiseux expansions of the integrand at infinity and n generalization of the residues to compute the integral. We also generalize a result of Rothstein that gives us a necessary condition for elementary integrability, and provide examples of its use.