Mathematics in Computer Science

, Volume 6, Issue 1, pp 79–105

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

Article

## 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

## References

1. 1.
Bartholomew-Biggs M.C.: Using forward accumulation for automatic differentiation of implicitly-defined functions. Comput. Optim. Appl. 9, 65–84 (1998)
2. 2.
Bender E.A., Bruce Richmond L.: A multivariate Lagrange inversion formula for asymptotic calculations. Electron. J. Comb 5, 1–4 (1998)Google Scholar
3. 3.
Bernardini A., Natalini P., Ricci P.E.: Multidimensional bell polynomials of higher order. Comput. Math. Appl. 0, 1–11 (2005)
4. 4.
Cayley, A.: Note on a differential equation. In: Manchester Society Proceedings, pp. 111–114 (1862)Google Scholar
5. 5.
Chou W.S., Hsu L.C., Shiue P.J.-S.: Application of Faa di Bruno’s formula in characterization of inverse relations. J. Comput. Appl. Math. 190, 151–169 (2006)
6. 6.
Christianson B.: Reverse accumulation and implicit functions. Optim. Methods Softw. 9, 307–322 (1998)
7. 7.
Constantine G.M., Savits T.H.: A multivariable Faa di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)
8. 8.
Craik A.D.D.: Prehistory of Faa di Bruno’s formula. Am. Math. Monthly. 112(2), 119–130 (2005)
9. 9.
Egorychev, G.P.: Integral Representation and the Computation of Combinatorial Sums. In: Translations of Mathematical Monographs, American Mathematical Society, vol. 59 (1984)Google Scholar
10. 10.
Faa di Bruno F.: Sullo sviluppo delle Funzioni. Annal di Scienze Mathematiche e Fisiche. 6, 479–480 (1855)Google Scholar
11. 11.
Faa di Bruno F.: Note sur une nouvelle formule de calcul differentiel. Q. J. Math. 1, 359–360 (1857)Google Scholar
12. 12.
Gallop, E.G.: Change of the independent variable in a differential coefficient. Trans. Camb. Phil. Soc. 16, 116–132Google Scholar
13. 13.
Hardy G.H., Ramanujan S.: Asymptotic formula in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)
14. 14.
Johnson W.P.: The curious history of Faa di Bruno’s formula. Am. Math. Monthly. 109(3), 217–234 (2002)
15. 15.
Kamber F.: Formules exprimant les valeurs des coefficients des series de puissances inverses. Acta Mathematica 78, 193–204 (1946)
16. 16.
Lagrange J.-L.: Nouvelle methode pour resoudre les equations litterales par le moyen des series. Memoires de l’Academie Royale des Sciences et Belles-Lettres de Berlin. 24, 251–326 (1770)Google Scholar
17. 17.
Ma T.-Y.: Higher chain formula proved by combinatorics. Electron. J. Comb 21, 1–7 (2009)Google Scholar
18. 18.
Mishkov R. L.: Generalization of the formula of Faa Di Bruno for a composite function with a vector argument. Int. J. Math. Math. Sci. 24(7), 481–491 (2000)
19. 19.
Mumford D.: Tata Lectures on Theta II. Birkhauser, Basel (1993)Google Scholar
20. 20.
Murphy R.: First memoir on the theory of analytical operations. Phil. Trans. Royal Soc. Lond. 127, 179–210 (1837)
21. 21.
Nahay, J.M.: Linear differential resolvents. Dissertation. Rutgers University, Piscataway (2000)Google Scholar
22. 22.
Scherk, H.F.: De evolvenda functione $${\frac{yd.yd.yd \ldots ydx}{dx^{n}}}$$ disquisitions nonnullae analyticae. Dissertation. Friedrich-Wilhelms-Universitat, Berlin (1823)Google Scholar
23. 23.
Sylvester J.J.: Note on Burman’s law for the inversion of the independent variable. Philosophical Magazine 8, 535–540 (1854) (4th series)Google Scholar
24. 24.
Wagner M., Schaefer B.-J., Walther A.: On the efficient computation of high-order derivatives for implicitly defined functions. Comput. Phys. Commun. 181, 756–764 (2010)