Related papers: A ridiculously simple and explicit implicit functi…
We give an explicit implicit function theorem for formal power series that is valid for all fields, which implies in particular Lagrange inversion formula and and Flajolet-Soria coefficient extraction formula known for fields of…
In this paper, we introduce a method of converting implicit equations to the usual forms of functions locally without differentiability. For a system of implicit equations which are equipped with continuous functions, if there are unique…
The phenomenon of an implicit function which solves a large set of second order partial differential equations obtainable from a variational principle is explicated by the introduction of a class of universal solutions to the equations…
The authors study the classical Lagrange inversion theorem--an antecedent of the modern implicit function theorem--in the smooth case. Examples are given to show that the result is sharp.
We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.
This work introduces a new inversion formula for analytical functions. It is simple, generally applicable and straightforward to use both in hand calculations and for symbolic machine processing. It is easier to apply than the traditional…
We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of…
The goal of the paper is to present two simple proofs of the Lagrange Inversion Formula for formal power series. Both proofs are non-external in the sense that they use concepts that do not go beyond the scope of formal power series…
We give an algorithm for reversion of formal power series, based on an efficient way to implement the Lagrange inversion formula. Our algorithm requires $O(n^{1/2}(M(n) + MM(n^{1/2})))$ operations where $M(n)$ and $MM(n)$ are the costs of…
We present a simple inductive proof of the Lagrange Inversion Formula.
This article presents simple and easy proofs of the Implicit Function Theorem and the Inverse Function Theorem, in this order, both of them on a finite-dimensional Euclidean space, that employ only the Intermediate Value Theorem and the…
This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…
The Implicit and Inverse Function Theorems are special cases of a general Implicit/Inverse Function Theorem which can be easily derived from either theorem. The theorems can thus be easily deduced from each other via the generalized…
We give presentation of composition inverse of formal power serie in a logarithmic form.
We formulate some global invertibility and implicit function theorems. We extend the result of Idczak, Skowron and Walczak on the invertibility of the operators to the case of the operators with critical points. The proof relies on the…
Sufficient conditions are given for a hard implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when…
We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the…
Starting from the representation of a function $f(x,y)$ as a formal power series with Taylor coefficients $f_{m,n}$, we establish a formal series for the implicit function $y=y(x)$ such that $f(x,y)=0$ and the coefficients of the series for…
We propose a calculus for modeling implicit programming that supports first-class, overlapping, locally scoped, and higher-order instances with higher-kinded types. We propose a straightforward generalization of the well-established System…
We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ…