Exercise 1, § 7

Let $g$ be a complete normed Lie algebra over $K$ such that

for $x, y$ in $g$. Let $\Theta$ denote the set of $x \in g$ such that $\|x\| < \frac{1}{3} \log \frac{3}{2}$. For $x, y$ in $\Theta$, $h(x, y) \in \Theta$, cf. no. 2.

(a) We write $f(T) = (1 - e^{-T})/T$ and $g(T) = 1/f(T)$. The series $f$ converges throughout the complex plane and the series $g$ on the open disc of radius $2\pi$ (cf. Functions of a Real Variable, Chapter VI, § 2, no. 3). Deduce that, if $z \in \Theta$, $f(\mathrm{ad}\, z)$ and $g(\mathrm{ad}\, z)$ are defined and that they are elements of $\mathscr{L}(g; g)$ and inverses of one another.

(b) Let $D_2 h(x, y)$ be the second partial derivative of $h$ at a point $(x, y)$ of $\Theta \times \Theta$ (Differentiable and Analytic Manifolds, R, 1.6.2). It is an element of $\mathscr{L}(g; g)$. Show that

(Use Exercise 3 (e) of § 6 to show this formula when $x$ and $y$ are sufficiently close to zero and pass from this to the general case by analytic continuation.)

Show that the formula

is valid at every point of the domain of absolute convergence of the formal power series $\tilde{H}$ (cf. Proposition 1).