初等函数积分的刘维尔定理Liouville's theorem on integration in terms of elementary functions

This talk should be regarded as an elementary introduction to differential algebra. It culminates in a purely algebraic proof, due to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of “elementary” integrals of “elementary” functions. The precise meaning of elementary will be specified. As an application of that theorem we prove that the indefinite integral ∫e^(x^2)dx cannot be expressed in terms of elementary functions. 这次演讲应该被看作是对微分代数的初步介绍。由M.Rosenlicht 于1835年提出的关于“初等”函数的“初等”积分的存在性的刘维尔定理,其结果是一个纯粹的代数证明。“基本”的精确意义将被指定。作为该定理的一个应用,我们证明了不定积分∫e^(x^2)dx不能用初等函数表示。