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 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)
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 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.
It is a well-known empirical result that differentiation, especially higher order differentiation, of simple expressions can lead to long and complex expressions. In this paper we give some theoretical results that help to explain this phenomenon. In particular we show that in certain representations there exist expressions whose representations require… (More)