Proposition 14

— We have $l(e(X)) = e(l(X)) = X$.

The series l and e have no constant term and their terms of degree 1 are equal to X. By Prop. 10 of IV, p. 37 it suffices to prove the formula $l(e(X)) = X$. By the formulae (36) and (37) and Cor. 3 of IV, p. 33 we have

whence $I(e(X)) = X$.

Let K be a Q-algebra, then the elements of K[[I]] without constant term form a commutative group $\mathscr{E}$ under addition. The elements of K[[I]] with constant term 1 form a commutative group $\mathbf{A}$ under multiplication (IV, p. 30). For each $f \in \mathscr{E}$, we can define the elements $e \circ f$ and $I \circ f$ of $\mathscr{E}$, and by Prop. 14 above, the mappings $f \mapsto l \circ f$ and $f \mapsto e \circ f$ are mutually inverse permutations of $\mathscr{E}$; clearly they are continuous. Since $\exp X = e(X) + 1$, we see that the exponential mapping $f \mapsto \exp f = e \circ f + 1$ is a continuous bijection of $\mathscr{E}$ onto $\mathscr{M}$. By formula (4) of IV, p. 29 and Prop. 13, we have $\exp(f+g) = (\exp f)(\exp g)$ for $f, g \in \mathscr{E}$. Thus the exponential is an isomorphism of the topological group $\mathscr{E}$ onto the topological group $\mathbf{A}$.

The inverse isomorphism of $\mathscr{M}$ onto $\mathscr{E}$ is called the logarithm and is written $g \mapsto \log g$. We thus have $\log g = l(g-1)$ for $g$ in $\mathscr{M}$, and in particular,

(38)

Since the logarithm is a homomorphism of $\mathscr{M}$ into $\mathscr{E}$, the formula $(1+X)(1+Y) = 1 + (X+Y+XY)$ implies

(39)

Let $(u_\lambda)_{\lambda \in L}$ be a summable family of elements of $\mathscr{B}$, then the family $(\exp u_{\lambda})_{\lambda \in L}$ is multipliable and we have

(40)

Similarly, if $(f_\lambda)_{\lambda \in L}$ is a multipliable family of elements of A, the family $(\log f_\lambda)_{\lambda \in L}$ is summable and we have

(41)

Let $g \in A$, and let D be a continuous derivation of $K[[I]]$. We have $\log g = l(g - 1)$, hence by Cor. 3 of IV, p. 33 and (37) we have

(42)

The expression $D(g)/g$ is called the logarithmic derivative of g (relative to D).