计算某种“闭形式”的不定积分,即符号积分,是计算机代数的一个重要研究领域。在部分实现递归Risch算法后,人们发现并行积分方法可以实现更高效的算法,其中最著名的算法之一是Risch-Norman算法。然而,这种方法依赖于积分中无法准确得到的多项式次数的估计。Norman基于完备化思想提供了一种避免次数估计的替代方法。然而,根据微分域的构造和项序的选择,可能会发生完备化过程不能终止并产生无限多约化法则的情况。我们将Norman方法优化并应用于在物理学中有重要应用的Airy函数生成的微分环。通过确定适当的项序,我们用有限个公式表示无限多个约化法则,并给出了Airy函数的两个完备约化系统。

The computation of indefinite integrals in certain kind of “closed form”,which is known as symbolic integration,is an important research subarea of computer algebra.After implementing the recursive Risch algorithm partly,it was realized that efficient algorithms can be achieved by a parallel approach.This led to the famous Risch-Norman algorithm.However,this approach relies on an ansatz with heuristic degree bounds.Norman’s completion-based approach provides an alternative for finding the numerator of the integral avoiding heuristic degree bounds.However,depending on the differential field and on the selected ordering of terms,it may happen that the completion process does not terminate and yields an infinite number of reduction rules.We apply Norman’s approach to the differential field generated by Airy functions,which play an important role in physics.By fixing adapted orderings and analyzing the process in the concrete case,we present two complete reduction systems for Airy functions by finitely many formulae to denote infinitely many reduction rules.