Prehistory of Faà di Bruno's Formula

@article{Craik2005PrehistoryOF,
  title={Prehistory of Fa{\`a} di Bruno's Formula},
  author={Alex D. D. Craik},
  journal={The American Mathematical Monthly},
  year={2005},
  volume={112},
  pages={119 - 130}
}
  • A. Craik
  • Published 1 February 2005
  • History
  • The American Mathematical Monthly
poraries on the series expansion of composite functions. Here, earlier work unnoticed in these papers is described. These anticipate many of the results attributed to others. This earlier work originates with Arbogast, with later reworkings by Knight, West, De Morgan, and others. The series expansion in powers of x of a suitably-differentiable composite function g(f(x)) has the form 
View on Taylor & Francis
jstor.org
80 Citations
Highly Influential Citations
4
Background Citations
27
Methods Citations
12

80 Citations

A new Faà di Bruno type formula

Faà di Bruno’s formula for the derivatives of composite functions has an interesting history and a rich literature (see, e.g., the survey by Johnson [8], the revealing paper on the predecessors of…

The multivariate Faà di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

Abstract The Faà di Bruno formulæ for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial literature on the univariate…

Five interpretations of Fa\`a di Bruno's formula

In these lectures we present five interpretations of the Fa' di Bruno formula which computes the n-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras…

Fa a di Bruno's formula for variational calculus

This paper determines the general formula for describing dierentials of composite functions in terms of dierentials of their factor functions. This generalises the formula commonly attributed to Fa a…

A discrete Faà di Bruno's formula

We derive a discrete Faa di Bruno's formula that rules the behaviour of finite differences under composition of functions with vector values and arguments.

Faà di Bruno Hopf algebras

This is a short review on the Faà di Bruno formulas, implementing composition of real-analytic functions, and a Hopf algebra associated to such formulas. This structure provides, among several other…

Faa di Bruno's formula for chain differentials

This paper determines the general formula for describing differentials of composite functions in terms of differentials of their factor functions. This generalises

On the fractional version of Leibniz rule

This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann–Liouville and Caputo fractional derivatives of order α∈(0,1) . In the context…

A one-sentence elementary proof of the combinatorial Fa\`a di Bruno's formula

Faà di Bruno’s formula generalizes the chain rule of elementary calculus. It tells us the nth derivative f(g(x)) of a composition of functions. See [3] and [1] for the remarkable history of this…

Formal differential variables and an abstract chain rule

. One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f ( g ( x )), but not (directly) the second or higher-order derivatives. We present iterated differentials…
...

26 References

The Curious History of Faà di Bruno's Formula

A restatement in terms of set partitions can be proved easily in a few lines, as the authors shall see in Section 2, though it still requires a bit of work to pass from that form to the form in (1.1).

On the expansion of any functions of multinomials

  • T. Knight
  • Mathematics
    Philosophical Transactions of the Royal Society of London
  • 1811
1. The expansion of multinomial functions has, of late, been so ably and fully treated by M. Arbogast, in his learned work ‘Du Calcul des Dérivations,' that it may appear, perhaps, scarcely necessary…

From Ford to Faà

A visual approach to the Chung-Feller Theorem and two uniformly distributed parameters defining Catalan numbers.

Calculus and Analysis in Early 19th-Century Britain: The Work of William Wallace

Abstract Scottish-born William Wallace (1768–1843) was an early exponent of the differential calculus in Britain and translator of French mathematical works. Encyclopaedias published during the early…

Geometry, Analysis, and the Baptism of Slaves: John West in Scotland and Jamaica

Abstract The achievements of the little-known Scottish mathematician, John West (1756–1817), deserve recognition: hisElements of Mathematics(1784) shows him to be a skilled expositor and innovative…

III. A method of raising an infinite multinomial to any given power, or extracting any given root of the same

  • A. D. Moivre
  • Mathematics
    Philosophical Transactions of the Royal Society of London
  • 1695
Tis about two Years Since, that considering Mr. Newton's Theorem for Raising a Binomial to any given Power, or Extracting any Root of the Same; I enquired, whether What he had done for a Binomial,…

The Art in Computer Programming

Here the authors haven’t even started the project yet, and already they’re forced to answer many questions: what will this thing be named, what directory will it be in, what type of module is it, how should it be compiled, and so on.

Du calcul des dérivations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Documents about the work Du calcul des dérivations (1800) / Louis-François-Antoine Arbogast (1759-1803) . . . . . . . . . . . .…

Theorie der independenten Darstellung der höheren Differentialquotienten

The generalized chain rule of differentiation with historical notes

  • Utilitatis Mathematica
  • 2002