词条 | Faà di Bruno's formula |
释义 |
Perhaps the most well-known form of Faà di Bruno's formula says that where the sum is over all n-tuples of nonnegative integers (m1, …, mn) satisfying the constraint Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit: Combining the terms with the same value of m1 + m2 + ... + mn = k and noticing that m j has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of Bell polynomials Bn,k(x1,...,xn−k+1): Combinatorial formThe formula has a "combinatorial" form: where
Explanation via an exampleThe combinatorial form may initially seem forbidding, so let us examine a concrete case, and see what the pattern is: The pattern is The factor corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor that goes with it corresponds to the fact that there are three summands in that partition. The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1. Similarly, the factor in the third line corresponds to the partition 2 + 2 of the integer 4, (4, because we are finding the fourth derivative), while corresponds to the fact that there are two summands (2 + 2) in that partition. The coefficient 3 corresponds to the fact that there are ways of partitioning 4 objects into groups of 2. The same concept applies to the others. A memorizable scheme is as follows: Combinatorics of the Faà di Bruno coefficientsThese partition-counting Faà di Bruno coefficients have a "closed-form" expression. The number of partitions of a set of size n corresponding to the integer partition of the integer n is equal to These coefficients also arise in the Bell polynomials, which are relevant to the study of cumulants. VariationsMultivariate versionLet y = g(x1, ..., xn). Then the following identity holds regardless of whether the n variables are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables (if it seems opaque, see the very concrete example below):[3] where (as above)
More general versions hold for cases where the all functions are vector- and even Banach-space-valued. In this case one needs to consider the Fréchet derivative or Gateaux derivative.
The five terms in the following expression correspond in the obvious way to the five partitions of the set { 1, 2, 3 }, and in each case the order of the derivative of f is the number of parts in the partition: If the three variables are indistinguishable from each other, then three of the five terms above are also indistinguishable from each other, and then we have the classic one-variable formula. ===Formal power series version=== Suppose and are formal power series and . Then the composition is again a formal power series, where c0 = a0 and the other coefficient cn for n ≥ 1 can be expressed as a sum over compositions of n or as an equivalent sum over partitions of n: where is the set of compositions of n with k denoting the number of parts, or where is the set of partitions of n into k parts, in frequency-of-parts form. The first form is obtained by picking out the coefficient of xn in "by inspection", and the second form is then obtained by collecting like terms, or alternatively, by applying the multinomial theorem. The special case f(x) = ex, g(x) = ∑n ≥ 1 an /n! xn gives the exponential formula. The special case f(x) = 1/(1 − x), g(x) = ∑n ≥ 1 (−an) xn gives an expression for the reciprocal of the formal power series ∑n ≥ 0 an xn in the case a0 = 1. Stanley [4] gives a version for exponential power series. In the formal power series we have the nth derivative at 0: This should not be construed as the value of a function, since these series are purely formal; there is no such thing as convergence or divergence in this context. If and and then the coefficient cn (which would be the nth derivative of h evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by where {{pi}} runs through the set of all partitions of the set {1, ..., n} and B1, ..., Bk are the blocks of the partition {{pi}}, and | Bj | is the number of members of the jth block, for j = 1, ..., k. This version of the formula is particularly well suited to the purposes of combinatorics. We can also write with respect to the notation above where Bn,k(a1,...,an−k+1) are Bell polynomials. A special caseIf f(x) = ex, then all of the derivatives of f are the same and are a factor common to every term. In case g(x) is a cumulant-generating function, then f(g(x)) is a moment-generating function, and the polynomial in various derivatives of g is the polynomial that expresses the moments as functions of the cumulants. Notes1. ^{{harv|Arbogast|1800}}. 2. ^According to {{harvtxt|Craik|2005|pp=120–122}}: see also the analysis of Arbogast's work by {{harvtxt|Johnson|2002|p=230}}. 3. ^{{cite journal |last=Hardy |first=Michael |title=Combinatorics of Partial Derivatives |journal=Electronic Journal of Combinatorics |volume=13 |issue=1 |year=2006 |pages=R1 |url=http://www.combinatorics.org/Volume_13/Abstracts/v13i1r1.html |ref=harv }} 4. ^See the "compositional formula" in Chapter 5 of {{cite book |last=Stanley |first=Richard P. |title=Enumerative Combinatorics |origyear=1997 |year=1999 |isbn=978-0-521-55309-4|publisher=Cambridge University Press |url=http://www-math.mit.edu/~rstan/ec/}} ReferencesHistorical surveys and essays
| last = Brigaglia | first = Aldo | editor-last = Giacardi | editor-first = Livia | contribution = L'Opera Matematica | title = Francesco Faà di Bruno. Ricerca scientifica insegnamento e divulgazione | place = Torino | language = Italian | publisher = Deputazione Subalpina di Storia Patria | year = 2004 | series = Studi e fonti per la storia dell'Università di Torino | volume = XII | pages = 111–172}}. "The mathematical work" is an essay on the mathematical activity, describing both the research and teaching activity of Francesco Faà di Bruno.
| first=Alex D. D. | last=Craik | title=Prehistory of Faà di Bruno's Formula | journal=American Mathematical Monthly | volume=112 | date=February 2005 | pages=217–234 | issue=2 | jstor = 30037410 | doi=10.2307/30037410 | mr= 2121322 | zbl= 1088.01008 }}.
| last = Johnson | first = Warren P. | title = The Curious History of Faà di Bruno's Formula | journal = American Mathematical Monthly | volume = 109 | issue = 3 | pages = 217–234 | date=March 2002 | url = https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Johnson217-234.pdf | jstor = 2695352 | doi = 10.2307/2695352 | mr = 1903577 | zbl = 1024.01010 | citeseerx = 10.1.1.109.4135 Research works
| first = L. F. A. | last=Arbogast | author-link= Louis François Antoine Arbogast | title=Du calcul des derivations | trans-title=On the calculus of derivatives | year=1800 | language=French | publisher=Levrault | place=Strasbourg | url=https://books.google.com/books?id=YoPq8uCy5Y8C | pages= xxiii+404 }}, Entirely freely available from Google books.
| first=F. | last=Faà di Bruno | author-link=Francesco Faà di Bruno | title= Sullo sviluppo delle funzioni | trans-title=On the development of the functions | language = Italian | journal= Annali di Scienze Matematiche e Fisiche | lccn=06036680 | volume= 6 | year= 1855 | pages= 479–480 | url = https://books.google.com/books?id=ddE3AAAAMAAJ&pg=PA479 }}. Entirely freely available from Google books. A well-known paper where Francesco Faà di Bruno presents the two versions of the formula that now bears his name, published in the journal founded by Barnaba Tortolini.
| first=F. | last=Faà di Bruno | title= Note sur une nouvelle formule de calcul differentiel | trans-title=On a new formula of differential calculus | language = French | journal= The Quarterly Journal of Pure and Applied Mathematics | volume= 1 | year=1857 | pages= 359–360 | url = https://books.google.com/books?id=7BELAAAAYAAJ&pg=PA359 }}. Entirely freely available from Google books.
| first = Francesco | last= Faà di Bruno | title=Théorie générale de l'élimination | trans-title=General elimination theory | year=1859 | language= French | publisher=Leiber et Faraguet | place=Paris | url=https://books.google.com/books?id=MZ0KAAAAYAAJ | pages= x+224 }}. Entirely freely available from Google books.
| last = Fraenkel | first = L. E. | title = Formulae for high derivatives of composite functions | journal = Mathematical Proceedings of the Cambridge Philosophical Society | volume = 83 | issue = 2 | pages = 159–165 | year = 1978 | url = http://journals.cambridge.org/abstract_S0305004100054402 | doi = 10.1017/S0305004100054402 | mr = 0486377 | zbl = 0388.46032 | subscription=yes |via=Cambridge University Press }}.
| last1 = Krantz | first1 = Steven G. | author1-link = Steven G. Krantz | last2 = Parks | first2 = Harold R. | author2-link = Harold R. Parks | title = A Primer of Real Analytic Functions | place = Boston | publisher = Birkhäuser Verlag | year = 2002 | series = Birkhäuser Advanced Texts - Basler Lehrbücher | edition = Second | pages = xiv+205 | url = https://books.google.com/books?id=i4vw2STJl2QC | mr = 1916029 | zbl = 1015.26030 | isbn = 978-0-8176-4264-8 }}
| last = Porteous | first = Ian R. | author-link = Ian Robertson Porteous | title = Geometric Differentiation | place = Cambridge | publisher = Cambridge University Press | year = 2001 | edition = Second | chapter = Paragraph 4.3: Faà di Bruno's formula | chapterurl = https://books.google.com/books?id=BNrW0UJ_UFcC&pg=PA83 | pages = 83–85 | url = https://books.google.com/books?id=BNrW0UJ_UFcC | mr = 1871900 | zbl = 1013.53001 | isbn = 978-0-521-00264-6 }}.
| last = T. A. | first = (Tiburce Abadie, J. F. C.) | title = Sur la différentiation des fonctions de fonctions | trans-title=On the derivation of functions | language = French | journal = Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale | series = Série 1 | volume = 9 | pages = 119–125 | year = 1850 | url = http://www.numdam.org/item?id=NAM_1850_1_9__119_1 }}, available at NUMDAM. This paper, according to {{harvtxt|Johnson|2002|p=228}} is one of the precursors of {{harvnb|Faà di Bruno|1855}}: note that the author signs only as "T.A.", and the attribution to J. F. C. Tiburce Abadie is due again to Johnson.
| last = A. | first = (Tiburce Abadie, J. F. C.) | title = Sur la différentiation des fonctions de fonctions. Séries de Burmann, de Lagrange, de Wronski | trans-title=On the derivation of functions. Burmann, Lagrange and Wronski series. | language = French | journal = Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale | series = Série 1 | volume = 11 | pages = 376–383 | year = 1852 | url = http://www.numdam.org/item?id=NAM_1852_1_11__376_1 }}, available at NUMDAM. This paper, according to {{harvtxt|Johnson|2002|p=228}} is one of the precursors of {{harvnb|Faà di Bruno|1855}}: note that the author signs only as "A.", and the attribution to J. F. C. Tiburce Abadie is due again to Johnson. External links
6 : Differentiation rules|Differential calculus|Differential algebra|Enumerative combinatorics|Factorial and binomial topics|Theorems in analysis |
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