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)
In this paper we discuss the problem of simplifying unnested radical expressions. We describe an algorithm implemented in MACSYMA that simplifies radical expressions and then follow this description with a formal treatment of the problem. Theoretical computing times for some of the algorithms are briefly discussed as is related work of other authors.
In this paper a generalization of the Risch Structure Theorem is reported. The generalization applies to fields F (t<inf>1</inf>,...,t<inf>n</inf>) where F is a differential field (in our applications F will be a finitely generated extension of Q, the field of rational numbers) and each t<inf>i</inf> is either algebraic over F<inf>i-1</inf> =… (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)
In this paper new algorithms are given for Gaussian integer division and the calculation of the greatest common divisor of two Gaussian integers. Empirical tests show that the new gcd algorithm is up to 5.39 times as fast as a Euclidean algorithm using the new division algorithm.
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)
To Bobby F. Caviness on the occasion of his 60th birthday 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 1960s,… (More)