TITLE: Genealogy of the Lagrange inversion theorem
QUESTION [27 upvotes]: 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 $f_{k}$)

REPLY [11 votes]: This theorem has been proven by many people and in the context of many diverse fields of mathematics since Lagrange presented "Nouvelle methode pour resoudre les equations litterales par le moyen des series."
It is so ubiiquitous that I've often thought that getting all the proofs together in one place would capture in a unique and compelling way the ebb and flow of the styles, habits, aesthetics and interests of the mathematics community over time. Indeed, I started on this project and quickly discovered that it was more than I could possibly do on my own. Just starting the project however gave me an even deeper appreciation of works such as Dickson's History of the Theory of Numbers and Fletcher's Index of Mathematical Tables.
I've got a list of 50+ proofs, a large collection of books, papers and articles and a bibtex bibliography if anybody is intrigued.  
IMHO, one of the most general statements is Krattenhaler's "Operator Methods and Lagrange Inversion: A Unified Approach to Lagrange Formulas."  One of the most unique is Abdesselam's "A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula."
Cheers, Scott (sbg at acw dot com)