Mathematics in Computer Science

, Volume 6, Issue 1, pp 79–105 | Cite as

The nth Order Implicit Differentiation Formula for Two Variables with an Application to Computing All Roots of a Transcendental Function

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Abstract

We prove a version of a generalization of the Lagrange inversion formula (LIF) for an implicit equation G(z, w) = 0 of two variables, expressing the nth derivative of z with respect to w as a polynomial in the mixed partial derivatives function of G with respect to z and w, and negative powers of the separant \({G_z \equiv \frac{\partial G}{\partial z}}\). Our method of proof is original, using only induction, and hence requires only that G be n-times differentiable in both variables, and requires only that the separant be nonzero. We then move on to a novel application of this LIF-like formula to derive a power series formula for each of the countably infinitely many roots of a pseudopolynomial—a finite sum of powers of a variable but allowing the powers to be any complex numbers.

Keywords

Lagrange inversion Implicit differentiation Transcendental 

Mathematics Subject Classification (2010)

70H03 34K32 34A09 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Broadway Performance SystemsColumbusUSA

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