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194 CHAPTER 7. CALCULUS
Notation 24 In (7.51), we write v 0 = p q , where p, q ∈ K[x, θ 1, . . . , θ n ], but are
not necessarily relatively prime.
From now on, until section 7.8.1, we will assume that the θ i are transcendental.
We can then infer the following algorithm for integrating expressions in
this setting.
Algorithm 37 (Parallel Risch [NM77, Bro07])
Input: L = K(x, θ 1 , . . . , θ n ) a purely transcendental differential field with constants
K, f ∈ L
Output: g an elementary integral of f, or a proof that there is no such one
satisfying the assumptions of steps (1)–(3).
(1) Decide candidate v 1 , . . . , v m (we may have too many)
(2) Decide a candidate q (which may be a multiple of the true value)
(3) Decide degree bounds for p (which may be too large),
i.e. n 0 . . . , n n such that
p =
∑n 0
∑n 1
i 0=0 i 1=0
· · ·
∑n n
i n=0
c i0,i 1,...,i n
x i0
(4) Clear denominators in the derivative of (7.51)
(5) Solve the resulting linear equations for the c i and c i0,i 1,...,i n
(c 0,...,0 is the the constant of integration, and is never determined)
(6) if there’s a solution
then reduce p/q to lowest terms and return the result
else “integral not found”
As explained in [Bro07], it is the decisions taken in steps (1)–(3) which mean
that this is not a guaranteed “integrate or prove unintegrable” algorithm. The
work of the previous sections allows partial solutions:
(1) Those v 1 , . . . , v m which depend on θ n
(2) A candidate q (which may be a multiple of the true value)
But the multiple is in K(x, θ 1 , . . . , θ n−1 )[θ n ], not in K[x, θ 1 , . . . , θ n ]
(3) A degree bound n n for p as a polynomial in θ n
7.8.1 The Parallel Approach: Algebraic Expressions
7.9 Definite Integration
We have shown (Example 11) that ∫ exp(−x 2 ) has no elementary expression,
i.e. that it is a new expression, which we called erf. However, ∫ ∞
(where
−∞ e−x2
we have attached a precise numerical meaning e x to exp(x)) is well-known to
be π, which is essentially another way of saying that erf(±∞) = ±1, and is
therefore a mater of giving numerical values to functions — see Chapter 8.
TO BE COMPLETEDMeijer etc.
n ∏
j=1
θ ij
j