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

摘要

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 º frac¶G¶z{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.