Lagrange inversion theorem

The Lagrange inversion theorem expresses coefficients of the compositional inverse of a power series using coefficients of the original series. It also yields formulas for derivatives and other functions of that inverse.

Formal power series: An infinite sequence of coefficients manipulated algebraically without requiring convergence. The theorem applies to power series formally, so analytic convergence need not be assumed.
Lagrange–Bürmann formula: A general form of Lagrange inversion that computes coefficients of functions composed with an implicitly defined series. This extension handles more than coefficients of the inverse itself.
Joseph-Louis Lagrange: An Italian-French mathematician who made major contributions to analysis, mechanics, and number theory. His work established the inversion result now bearing his name.
Series reversion: The process of finding a power series that reverses another under composition. Direct reversion computes the inverse successively, while Lagrange inversion gives coefficients by formula.
Compositional inverse: A function or series that reverses another under composition. The theorem calculates coefficients of this inverse without constructing it directly.
Catalan number: The integer sequence counting structures such as binary trees, Dyck paths, and polygon triangulations. Lagrange inversion extracts its generating-function coefficients from an implicit equation.
Augustin-Louis Cauchy: A French mathematician who helped establish rigorous methods in analysis and complex function theory. His residue methods provide a standard proof of the inversion formula.
Newton's method: An iterative numerical method for approximating roots of equations using tangent lines. It approximates solutions numerically rather than extracting exact inverse-series coefficients.
Residue theorem: A theorem evaluating contour integrals through residues at singularities. A residue change of variables gives a compact proof of the coefficient formula.
Tree enumeration: The counting of trees and related structures by size or other parameters. Implicit generating functions for rooted trees often yield coefficients through inversion.

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Abel's binomial theoremRelated: Abel-type identities arise in coefficient calculations related to inversion formulas.