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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)
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)
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.
In this paper the Brown-Collins modular greatest common divisor algorithm for polynomials in Z[x<subscrpt>1</subscrpt>,...,x<subscrpt>v</subscrpt>], where Z denotes the ring of rational integers, is generalized to apply to polynomials in G[x<subscrpt>1</subscrpt>,...,x<subscrpt>v</subscrpt>], where G denotes the ring of Gaussian integers, i.e., complex… (More)
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)
Recently John Fitch [FITC 74] gave an algorithm for computing [A<sup>1/n</sup>] where A and n are positive integers. His algorithm, based on the classical Newton method, has the virtue of not only being an extremely simple algorithm but it is also fast - both in practice and in theory. One can hardly hope to improve upon an algorithm with such virtues. In… (More)