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7.3. THEORY: LIOUVILLE’S THEOREM 177
(c) θ is a logarithm over K, i.e. there is an η in K such that θ ′ = η ′ /η, which
is only an algebraic way of saying that θ = log η.
In the light of this definition, (7.26) would be interpreted as saying
−i
∫
1
x 2 + 1 dx = i 2 (θ 1 − θ 2 ) ,
i
1+ix .
(7.26 restated)
where θ 1 ′ =
1−ix and θ′ 2 =
We should note that, if θ is a logarithm of η, then so is θ +c for any constant
(Definition 81) c. Similarly, if θ is an exponential of η, so is cθ for any constant
c, including the case c = 0, which explains the stipulation of nonzeroness in
Definition 82(b).
A consequence of these definitions is that log and exp satisfy the usual laws
“up to constants”.
log Suppose θ i is a logarithm of η i . Then
(θ 1 + θ 2 ) ′ = θ ′ 1 + θ ′ 2
= η′ 1
η 1
+ η′ 2
η 2
= η′ 1η 2 + η 1 η ′ 2
η 1 η 2
= (η 1η 2 ) ′
η 1 η 2
,
and hence θ 1 + θ 2 is a logarithm of η 1 η 2 , a rule normally expressed as
log η 1 + log η 2 = log(η 1 η 2 ). (7.27)
Similarly θ 1 − θ 2 is a logarithm of η 1 /η 2 and nθ 1 is a logarithm of η n 1 (for
n ∈ Z: we have attached no algebraic meaning to arbitrary powers).
exp Suppose now that θ i is an exponential of η i . Then
(θ 1 θ 2 ) ′ = θ 1θ ′ 2 + θ 1 θ 2
′
= η 1θ ′ 1 θ 2 + θ 1 η 2θ ′ 2
= (η 1 + η 2 ) ′ (θ 1 θ 2 )
and hence θ 1 θ 2 is an exponential of η 1 + η 2 , a rule normally expressed as
exp η 1 exp η 2 = exp(η 1 + η 2 ). (7.28)
Similarly θ 1 /θ 2 is an exponential of η 1 −η 2 and θ n 1 is an exponential of nη 1
(for n ∈ Z: we have attached no algebraic meaning to arbitrary powers).