A new symbolic computation for formal integration with exact power series. (English) Zbl 1076.65021
The authors propose a symbolic algorithm to compute exact power series solutions of integrals in the form:
\[
\int \frac{e^{u(x)}} {[f(x)]^2}\,dx
\]
where \(f(x)\) is a solution of the second order linear homogeneous differential equation:
\[
a_0(x) \frac{d^2y} {dx^2}+a_1(x) \frac{dy}{dx}+ a_2(x)y=0
\]
with \(a_0(x)\) and \(a_1(x)\) polynomial coefficients and
\[
u(x)=-\int \frac{a_1(x)} {a_0(x)}\,dx.
\]
They use algorithmic techniques and generalized hypergeometric series. Finally a Maple code is proposed.
Reviewer: Catterina Dagnino (Torino)
MSC:
| 65D30 | Numerical integration |
| 68W30 | Symbolic computation and algebraic computation |
| 30B10 | Power series (including lacunary series) in one complex variable |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
References:
| [1] | Bronstein, M., Symbolic Integration I (1997), Springer · Zbl 0880.12005 |
| [2] | D. Gruntz, Powerseries, a package for infinite power series, Maple Share Library, 1992; D. Gruntz, Powerseries, a package for infinite power series, Maple Share Library, 1992 |
| [3] | Redfern, D., The Maple Handbook (1994), Springer-Verlag · Zbl 0820.68002 |
| [4] | Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0695.33001 |
| [5] | Kıymaz, O.; Mirasyedioğlu, Ş., An algorithmic approach to exact power series solutions of second order linear homogeneous differential equations with polynomial coefficients, Appl. Math. Comput, 139, 165-178 (2003) · Zbl 1030.65075 |
| [6] | Koepf, W., Power series in computer algebra, J. Symb. Comput, 13, 581-603 (1992) · Zbl 0758.30026 |
| [7] | Koepf, W., Examples for the algorithmic calculation of formal Puiseux, Laurent and power series, SIGSAM Bull, 27, 20-32 (1993) |
| [8] | Ross, S. L., Differential Equations (1964), Ginn-Blaisdell: Ginn-Blaisdell London · Zbl 0151.10601 |
| [9] | Siret, Y., Computer Algebra (1993), Academic Press: Academic Press London · Zbl 0865.68064 |
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