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7.3. THEORY: LIOUVILLE’S THEOREM 181
7.3.3 Risch Structure Theorem
[Ris79] TO BE COMPLETED
7.3.4 Overview of Integration
Notation 23 Throughout sections 7.4–7.7, we assume that we are given an
integrand f ∈ L = C(x, θ 1 , . . . , θ n ), where L const = C is an effective field of constants,
each θ i is an elementary generator (Definition 82) over C(x, θ 1 , . . . , θ i−1 )
and, if θ i in given as an exponential or logarithmic generator, then θ i is actually
transcendental over C(x, θ 1 , . . . , θ i−1 ).
Ideally, the solution of the integration problem would then proceed by induction
on n: we assume we can integrate in K = C(x, θ 1 , . . . , θ n−1 ), and reduce
integration in L to problems of integration in K. It was the genius of Risch
[Ris69] to realise that a more sophisticated approach is necessary.
Definition 84 For a differential field L, the elementary integration problem for
L is to find an algorithm which, given an element f of L, finds an elementary
overfield M of L, and g ∈ M with g ′ = f, or proves that such M and g do not
exist.
Definition 85 For a differential field L, the elementary Risch differential equation
problem for L is to find an algorithm which, given elements f and g of L
(with f such that exp( ∫ f) is a genuinely new expression, i.e., for any non-zero
F with F ′ = fF , M = L(F ) is transcendental over L and has M const = L const ),
finds y ∈ L with y ′ + fy = g, or proves that such a y does not exist. We write
RDE(f, g) for the solution to this problem.
The reader might object “but I can solve this by integrating factors!”. Indeed,
if we write y = z exp( ∫ −f), we get
∫
z ′ exp(
( ∫
z exp(
−f) − fz exp(
′ ∫
−f))
+ fz exp(
∫
∫
−f) + fz exp(
−f) = g
−f) = g
which simplifies to
∫
z ′ exp(
−f) = g
and hence z ′ = g exp( ∫ f), z = ∫ g exp( ∫ f) and
∫ ∫ ∫
y = exp( −f) g exp( f). (7.31)
However, if we come to apply the theory of section 7.5 to the principal integral
(marked ∫ ) of (7.31), we find the integration problem reduces to solving the
∗
original y ′ + fy = g. Hence this problem must be attacked in its own right,
which we do in section 7.7 for the base case f, g ∈ C(x).
∗