Séminaire Lotharingien de Combinatoire, B61Ad (2009), 21 pp.

Alan D. Sokal

A Ridiculously Simple and Explicit Implicit Function Theorem

Abstract. I show that the general implicit-function problem (or parametrized fixed-point problem) in one complex variable has an explicit series solution given by a trivial generalization of the Lagrange inversion formula. I give versions of this formula for both analytic functions and formal power series.

Received: January 31, 2009. Accepted: July 14, 2009. Final Version: July 15, 2009.

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