A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, is the Lagrange inversion theorem for power series (even in the formal context).
Starting from the exact sequence
$0 \rightarrow \mathbb{C} \rightarrow \mathbb{C}((z)) \xrightarrow{D} \mathbb{C}((z)) \;\xrightarrow{ \mathrm{Res} }\; \mathbb{C} \rightarrow 0,$
and using the simple rules of the formal derivative D, and of the formal residue Res (that by definition is the linear form taking the formal Laurent series $f(z)=\sum_{k=m}^{\infty}f_k z^k \in \mathbb{C}((z))$ to the coefficient $f_{-1}$ of $z^{-1}$) one easily proves:
(Lagrange inversion formula): If $f(z):=\sum_{k=1}^{\infty}f_k z^k\in \mathbb{C}[[z]]$ and $g(z):=\sum_{k=1}^{\infty}g_k z^k\in\mathbb{C}[[z]]$ are composition inverse of each other, the coefficients of the (multiplicative) powers of $f$ and $g$ are linked by the formula $$n[z^n]g^k=k[z^{-k}]f^{-n},$$ and in particular (for $k=1$), $$[z^n]g=\frac{1}{n} \mathrm{Res}( f^{-n} ).$$
(to whom didn't know it: enjoy computing the power series expansion of the composition inverse of $f(z):=z+z^m$, or of $f(z):=z\exp(z),$ and of their powers).
My question: what are the generalization of this theorem in wider context. I mean, in the same way that, just to make one example, the archetypal ODE $u'=\lambda u$ procreates the theory of semigroups of evolution in Banach spaces.
Also, I'd be grateful to learn some new nice application of this classic theorem.
(notation: for $f=\sum_{k=m}^{\infty}f_k z^k \in \mathbb{C}((z))$ the symbol $[z^k]f$ stands, of course, for the coefficient $f_{k}$)
