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In this paper we give an extension of the Liouville theorem [RISC69, p. 169] 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,(More)
This paper deals with the simplification problem of symbolic mathematics. The notion of canonical form is defined and presented as a well-defined alternative to the concept of simplified form. Following Richardson it is shown that canonical forms do not exist for sufficiently rich classes of mathematical expressions. However, with the aid of a(More)
The success of the symbolic mathematical computation discipline is striking. The theoretical advances have been continuous and significant: Gröbner bases, the Risch integration algorithm, integer lattice basis reduction, hypergeometric summation algorithms, etc. From the beginning in the early 60s, it has been the tradition of our discipline to create(More)
This paper gives a corollary to Schanuel's conjecture that indicates when an exponential or logarithmic constant is transcendental over a given field of constants. The given field is presumed to have been built up by starting with the rationals Q with π adjoined and taking algebraic closure, adjoining values of the exponential function or of some fixed(More)
We show that in the ring generated by the integers and the functions x, sin x n and sin(x · sin x n) (n = 1, 2,. . .) defined on R it is undecidable whether or not a function has a positive value or has a root. We also prove that the existential theory of the exponential field C is undecidable. 1. Let S denote the class of expressions generated by the(More)
This paper uses elementary algebraic methods to obtain new proofs for theorems on algebraic relationships between the logarithmic and exponential functions. The main result is a multivariate version of a special case of the structure theorem due to Risch that gives, in a very explicit fashion, the possible algebraic relationships between the exponential and(More)