Integration in finite terms

Abstract

The first remark that must be made about integration in finite terms is that all the algorithms, and nearly all the implementations (Wang [1971] is the exception), deal with indefinite integration. Definite integration is largely restricted to a combination of table look-up and "guess the contour" heuristics. As has often been pointed out [Askey, 1984], definite integration is far more applicable than indefinite integration, since many special functions (β and Γ functions, for example) are defined by definite integrals, and whole tools, such as Laplace transforms, are built on definite integrals. We note that these problems are fundamentally algebraic, rather than numerical, since we are computing, not a number, but a function defined by F ( y ) = ∫ f ( x , y )d x . Very little is known about algorithmic definite integration, though recent work in transcendence theory (see, e.g., Chudnovsky [1982]) may at least provide some methods for showing negative results.