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 $f\_{k}$)



Genealogy of the Lagrange inversion theorem
The Lagrange inversion formula allows to give a combinatorial interpretation of the Jacobian conjecture. See for example the classic paper of Bass, Connell and Write, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, <http://www.ams.org/journals/bull/1982-07-02/S0273-0979-1982-15032-7/S0273-0979-1982-15032-7.pdf>.


