From: "J�rgen Will" Subject: Re: Closed formula Date: Sat, 23 Oct 1999 19:35:42 +0200 Newsgroups: sci.math Keywords: formal solutions to differential equations, summation, etc. Hankel O'Fung wrote: ... I think closed form solution means an elementary function solution. An elementary function is a "writable" function, writable only with known terms. Each elementary function is described by (complex) constants, polynomials, log and exp and their repeated compositions. It is formed by a differential field (see differential algebra: Ritt, Kolchin). There are decision procedures for some problems, which say if the problem has a closed form solution or not. For instance: Michael F. Singer: Formal Solutions of Differential Equations; J. Symb. Comput. 10 (1990) 59-94 Yiu-Kwong Man: On Computing Closed Forms for Indefinite Summations; J.Symb. Comput. 16 (1993) 355-376 R. Moenck: On Computing Closed Forms for Summation. MACSYMA (1977) 225-236 R.H. Risch: Algebraic Properties of Elementary Functions of Analysis; Am. J. Math. (1979) 101 Liouville�s Theorem Risch�s Algorithm.