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 
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.
Faa 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 allows, 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
Faa di Bruno's formula for Gateaux differentials and interacting stochastic population processes
The problem of estimating interacting systems of multiple objects is important to a number of different fields of mathematics, physics, and engineering. Drawing from a range of disciplines, including
...
...

References

SHOWING 1-10 OF 32 REFERENCES
The Curious History of Faà di Bruno's Formula
TLDR
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
TLDR
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
TLDR
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.
A method of raising an infinite multinomial to any given power, or extracting any given root of the same, Philos
  • Trans. R. Soc. London 19, No. 230 (1697) 619–625; also in Miscellanea analytica de seriebus et quadraturis, J. Tonson & J. Watts, London,
  • 1730
The generalized chain rule of differentiation with historical notes
  • Utilitatis Mathematica
  • 2002
...
...