B. F. Caviness

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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 wi th the simplification problem of symbolic mathemat ics . The notion of canonical form is defined and presented as a well-defined al ternat ive to the concept of simplified form. Following Richardson it is shown tha t canonical forms do not exist for sufficiently rich classes of mathemat ica l expressions. However, wi th the aid of a(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)
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, sinxn and sin(x · sinxn) (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 rational(More)
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)