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Chapter 7
Calculus
Throughout this chapter we shall assume that we are in characteristic zero, and
therefore all rings contains Z, and all fields contain Q. The emphasis in this
chapter will be on algorithms for integration. Historically, the earliest attempts
at integration in computer algebra [Sla61] were based on rules, and attempts to
“do it like a human would”. These were rapidly replaced by algorithmic methods,
based on the systematisation, in [Ris69], of work going back to Liouville.
Recently, attempts to get ‘neat’ answers have revived interest in rule-based approaches,
which can be surprisingly powerful on known integrals — see [JR10]
for one recent approach. In general, though, only the algorithmic approach is
capable of proving that an expression in unintegrable.
7.1 Introduction
We defined (Definition 31) the formal derivative of a polynomial purely algebraically,
and observed (Proposition 2.3.5) that it satisfied the sum and product
laws. We can actually make the whole theory of differentiation algebraic as follows.
Definition 80 A differential ring is a ring (Definition 6) equipped with an
additional unary operation, referred to as differentiation and normally written
with a postfix ′ , which satisfies two additional laws:
1. (f + g) ′ = f ′ + g ′ ;
2. (fg) ′ = fg ′ + f ′ g.
A differential ring which is also a field (Defintion 13) is referred to as a differential
field.
Definition 31 and Proposition 2.3.5 can then be restated as the following result.
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