Symbolic integration: the stormy decade
Communications of the ACM
The integration of a class of special functions with the Risch algorithm
ACM SIGSAM Bulletin
An approach to automatic asymptotic expansions
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
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In this presentation we describe the outline of an algorithmic approach to handle a class of algebraic integrands. (It is important to stress that for an extended abstract of the present form, we can at best convey the flavor of the approach, with numerous details missing.) We shall label this approach Carlson's algorithm because it is based on a series of analyses rendered by Carlson and his associates in the last ten years (Refs. 2, 3, 4, 8, and 12). The class of integrands is of the form r(x, y), where y2 is a polynomial in x, and r a rational function in x and y. This is the type of integrand that classically led to the study of elliptic integrals. At first glance this is a rather restricted class of algebraic functions. But in fact many trigonometric and hyperbolic integrands reduce to this form. The richness of this class of integrands is exemplified by a recently published handbook of 3000 integral formulas (Ref. 1). Our proposed approach will cover fifty to seventy percent of the items in the handbook. Furthermore the non-classical approach we shall describe holds great promise of developing to the case where definite integrals can be evaluated in terms of a host of other well-known functions (e.g., Bessel and Legendre).