An Extension of Liouville’s Theorem on Integration in Finite Terms

Abstract

Abstract. In Part of this paper, we give an extension of Liouville's Theorem and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone. Our main result generalizes Liouville's Theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral (but not the dilogorithm or exponential integral) to appear in the integral of an elementary function. The basic conclusion is that these functions, if they appear, appear linearly. We give an algorithm which decides if an elementary function, built up using only exponential functions and rational operations has an integral which can be expressed in terms of elementary functions and error functions.

Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients

“…If E is an elementary extension of C(x), the result follows from Risch (1968). If E is a purely transcendental liouvillian extension of C(x), the result follows from Theorem Al(b) of Singer et al (1985) and the fact that we can effectively embed such an extension in a log-explicit extension. We now assume that t'/t e E and let t'/t= v.…”

“…(ii) Either ~ v is in E or it is transcendental over E. Lemma 3.4 of Rothstein & Caviness (1979) and Theorem A1 of Singer et al (1985) imply that one can effectively embed E(t) into a regular (i.e. purely transcendental) log-explicit extension F of C. Furthermore F will be of the form E (t~,..., t,), with the t~ in E. The corollary to Theorem I of Rosenlicht (1976) implies that t is transcendental over F. Given u in E it is enough to decide if y'+ uy = 0 has a solution in F(t), since the corollary to Theorem 1 of Rosenlicht (1976) will imply this solution lies in E. Therefore, let us assume that E is a regular tog-explicit extension of C. Theorem Al(b) now allows us to decide if y'+ uy = 0 has a solution in E(t) and find such a solution if it does.…”

“…In this proof we shall rely heavily on the results of Rothstein & Caviness (1979) and the appendix of Singer et al (1985). If t'e E, then the Corollary to Theorem 1 of Rosenlicht (1976) implies that any solution u of y'+uy =0 in E(t) is actually in/3.…”

Structural theorems for symbolic summation

“…For an extensive list of literature and generalizations/refinements, like e.g. [50], we refer to [12].…”

Effective Set Membership in Computer Algebra and Beyond

“…We refer the reader to [3] for the standard definitions, and we let I be some class of functions (elementary, Liouvillian, EL [14] etc.). When we say "given an I function", we mean that it is given effectively, i.e.…”