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178 CHAPTER 7. CALCULUS
(1) Suppose θ is a logarithm of η, and φ is an exponential of θ. Then
φ ′ = θ ′ φ = η′
η φ,
φ ′
= η′
φ η = θ′ ,
and θ is a logarithm of φ, as well as φ being an exponential of η.
(2) Suppose now that θ is an exponential of η, and φ is a logarithm of θ. Then
φ ′ = θ′
θ
= η′ θ
θ = η′ ,
so η and φ differ by a constant. But φ, being a logarithm, is only defined
up to a constant.
(1)+(2) These can be summarised by saying that, up to constants, log and
exp are inverses of each other.
Definition 83 Let K be a field of expressions. An overfield K(θ 1 , . . . , θ n ) of
K is called a field of elementary expressions over K if every θ i is an elementary
generator over K(θ 1 , . . . , θ i−1 ). A expression is elementary over K if it belongs
to a field of elementary expressions over K.
If K is omitted, we understand C(x): the field of rational expressions.
For example, the expression exp(exp x) can be written as elementary over K =
Q(x) by writing it as θ 2 ∈ K(θ 1 , θ 2 ) where θ ′ 1 = θ 1 , so θ 1 is elementary over K,
and θ ′ 2 = θ ′ 1θ 2 , and so θ 2 is elementary over K(θ 1 ).
Observation 13 Other functions can be written this way as well. For example,
if θ ′ = iθ (where i 2 = −1), then( φ = 1 2i
(θ − 1/θ) is a suitable model for sin(x),
as in the traditional sin x = 1 2i e ix − e −ix) . Note that φ ′′ = −φ, as we would
hope.
From this point of view, the problem of integration, at least of elementary
expressions, can be seen as an exercise in the following paradigm.
Algorithm 32 (Integration Paradigm)
Input: an elementary expression f in x
Output: An elementary g with g ′ = f, or failure
Find fields C of constants, L of elementary expressions over C(x) with f ∈ L
if this fails
then error "integral not elementary"
Find an elementary overfield M of L, and g ∈ M with g ′ = f
if this fails
then error "integral not elementary"
else return g
so