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| {{Calculus |Differential}}
| | I like Fencing. Appears boring? Not!<br>I also to learn Norwegian in my spare time.<br><br>Also visit my web site ... [http://www.alleyboy.com/forum/hostgator-flask-350456 Hostgator Discount Coupons] |
| '''Faà di Bruno's formula''' is an identity in [[mathematics]] generalizing the [[chain rule]] to higher derivatives, named after {{harvs|txt|authorlink=Francesco Faà di Bruno|first=Francesco|last= Faà di Bruno|year=1855|year2=1857}}, though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician [[Louis François Antoine Arbogast]] stated the formula in a calculus textbook,<ref>{{cite book |first=L.F.A.|last=Arbogast|title=Du calcul des derivations |year=1800|publisher=Levrault|place=Strasbourg}}</ref> considered the first published reference on the subject.<ref>{{cite journal |first=A.D.D. |last=Craik |title=Prehistory of Faà di Bruno's Formula |journal=[[American Mathematical Monthly]] |volume=112 |year=2005 |pages=217–234|issue=2 |doi=10.2307/30037410 |publisher=Mathematical Association of America |ref=harv |postscript=. |jstor=30037410}}</ref>
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| Perhaps the most well-known form of Faà di Bruno's formula says that
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| :<math>{d^n \over dx^n} f(g(x))=\sum \frac{n!}{m_1!\,1!^{m_1}\,m_2!\,2!^{m_2}\,\cdots\,m_n!\,n!^{m_n}}\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(g^{(j)}(x)\right)^{m_j},</math>
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| where the sum is over all ''n''-[[tuple]]s of nonnegative integers (''m''<sub>1</sub>, …, ''m''<sub>''n''</sub>) satisfying the constraint
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| :<math>1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+n\cdot m_n=n.\,</math>
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| 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:
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| :<math>{d^n \over dx^n} f(g(x))
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| =\sum \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!}\cdot
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| f^{(m_1+\cdots+m_n)}(g(x))\cdot
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| \prod_{j=1}^n\left(\frac{g^{(j)}(x)}{j!}\right)^{m_j}.</math>
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| Combining the terms with the same value of ''m''<sub>1</sub> + ''m''<sub>2</sub> + ... + ''m''<sub>''n''</sub> = ''k'' and noticing that ''m''<sub> ''j''</sub> has to be zero for ''j'' > ''n'' − ''k'' + 1 leads to a somewhat simpler formula expressed in terms of [[Bell polynomial]]s ''B''<sub>''n'',''k''</sub>(''x''<sub>1</sub>,...,''x''<sub>''n''−''k''+1</sub>):
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| :<math>{d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).</math>
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| == Combinatorial form ==
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| The formula has a "combinatorial" form:
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| :<math>{d^n \over dx^n} f(g(x))=(f\circ g)^{(n)}(x)=\sum_{\pi\in\Pi} f^{(\left|\pi\right|)}(g(x))\cdot\prod_{B\in\pi}g^{(\left|B\right|)}(x)</math>
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| where
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| *π runs through the set Π of all [[partition of a set|partitions of the set]] { 1, ..., ''n'' },
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| *"''B'' ∈ π" means the variable ''B'' runs through the list of all of the "blocks" of the partition π, and
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| *|''A''| denotes the cardinality of the set ''A'' (so that |π| is the number of blocks in the partition π and |''B''| is the size of the block ''B'').
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| ==Explication via an example==
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| The combinatorial form may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:
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| :<math>
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| \begin{align}
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| (f\circ g)''''(x)
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| & = f''''(g(x))g'(x)^4
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| + 6f'''(g(x))g''(x)g'(x)^2 \\[8pt]
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| & {} \quad+\; 3f''(g(x))g''(x)^2
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| + 4f''(g(x))g'''(x)g'(x) \\[8pt]
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| & {} \quad+\; f'(g(x))g''''(x).
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| \end{align}
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| </math>
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| The pattern is
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| :<math>
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| \begin{align}
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| g'(x)^4
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| & & \leftrightarrow & & 1+1+1+1
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| & & \leftrightarrow & & f''''(g(x))
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| & & \leftrightarrow & & 1
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| \\[12pt]
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| g''(x)g'(x)^2
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| & & \leftrightarrow & & 2+1+1
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| & & \leftrightarrow & & f'''(g(x))
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| & & \leftrightarrow & & 6
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| \\[12pt]
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| g''(x)^2
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| & & \leftrightarrow & & 2+2
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| & & \leftrightarrow & & f''(g(x))
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| & & \leftrightarrow & & 3
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| \\[12pt]
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| g'''(x)g'(x)
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| & & \leftrightarrow & & 3+1
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| & & \leftrightarrow & & f''(g(x))
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| & & \leftrightarrow & & 4
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| \\[12pt]
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| g''''(x)
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| & & \leftrightarrow & & 4
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| & & \leftrightarrow & & f'(g(x))
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| & & \leftrightarrow & & 1.
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| \end{align}
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| </math>
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| [[Image:A memorizable pattern for the Faa da Bruno-formula.png|thumb|300px|A memorizable scheme]]
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| The factor <math>\scriptstyle g''(x)g'(x)^2 \;</math> corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor <math>\scriptstyle f'''(g(x))\;</math> 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.
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| Similarly, the factor <math>\scriptstyle g''(x)^2 \;</math> in the third line corresponds to the partition 2 + 2 of the integer 4, (4, because we are finding the fourth derivative), while <math>\scriptstyle f''(g(x)) \,\!</math> corresponds to the fact that there are ''two'' summands (2 + 2) in that partition. The coefficient 3 corresponds to the fact that there are <math>\tfrac{1}{2}\tbinom{4}{2}=3</math> ways of partitioning 4 objects into groups of 2. The same concept applies to the others.
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| ==Combinatorics of the Faà di Bruno coefficients==
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| These partition-counting '''Faà di Bruno coefficients''' have a "closed-form" expression. The number of [[partition of a set|partitions of a set]] of size ''n'' corresponding to the [[integer partition]]
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| :<math>\displaystyle n=\underbrace{1+\cdots+1}_{m_1}
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| \,+\, \underbrace{2+\cdots+2}_{m_2}
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| \,+\, \underbrace{3+\cdots+3}_{m_3}+\cdots</math>
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| <!-- Apparently \displaystyle is needed to make the subscripts on underbraces appear in the right position. -->
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| of the integer ''n'' is equal to
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| :<math>\frac{n!}{m_1!\,m_2!\,m_3!\,\cdots 1!^{m_1}\,2!^{m_2}\,3!^{m_3}\,\cdots}.</math>
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| These coefficients also arise in the [[Bell polynomials]], which are relevant to the study of [[cumulant]]s.
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| ==Variations==
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| ===Multivariate version===
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| Let ''y'' = ''g''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>). 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):<ref>{{cite journal |author=Hardy, 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 |postscript=.}}</ref>
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| :<math>{\partial^n \over \partial x_1 \cdots \partial x_n}f(y)
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| = \sum_{\pi\in\Pi} f^{(\left|\pi\right|)}(y)\cdot\prod_{B\in\pi}
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| {\partial^{\left|B\right|}y \over \prod_{j\in B} \partial x_j}</math>
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| where (as above)
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| *π runs through the set Π of all [[partition of a set|partitions of the set]] { 1, ..., ''n'' },
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| *"''B'' ∈ π" means the variable ''B'' runs through the list of all of the "blocks" of the partition π, and
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| *|''A''| denotes the cardinality of the set ''A'' (so that |π| is the number of blocks in the partition π and |''B''| is the size of the block ''B'').
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| A further generalization, due to Tsoy-Wo Ma, considers the case where ''y'' is a vector-valued variable.<ref>{{cite journal |author=Ma, Tsoy Wo|title=Higher Chain Formula proved by Combinatorics |journal=Electronic Journal of Combinatorics |volume=16 |issue=1 |year=2009 |pages=N21 |url=http://www.combinatorics.org/Volume_16/Abstracts/v16i1n21.html |ref=harv |postscript=.}}</ref>
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| The general form, for variational calculus ([[Gâteaux derivative|Gâteaux differentials]] are the most general form of differential), was derived in 2012.<ref>{{cite journal |author=Clark, Daniel and Houssineau, Jeremie |title=Hierarchical stochastic population processes |journal= |arxiv =1202.0264 |year=2012 |url=http://arxiv.org/abs/1202.0264 }}</ref>
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| ; Example
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| 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:
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| :<math>
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| \begin{align}
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| {\partial^3 \over \partial x_1\, \partial x_2\, \partial x_3}f(y)
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| & = f'(y){\partial^3 y \over \partial x_1\, \partial x_2\, \partial x_3} \\[10pt]
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| & {} + f''(y) \left( {\partial y \over \partial x_1}
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| \cdot{\partial^2 y \over \partial x_2\, \partial x_3}
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| +{\partial y \over \partial x_2}
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| \cdot{\partial^2 y \over \partial x_1\, \partial x_3}
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| + {\partial y \over \partial x_3}
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| \cdot{\partial^2 y \over \partial x_1\, \partial x_2}\right) \\[10pt]
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| & {} + f'''(y) {\partial y \over \partial x_1}
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| \cdot{\partial y \over \partial x_2}
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| \cdot{\partial y \over \partial x_3}.
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| \end{align}
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| </math> | |
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| 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.
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| ===Formal power series version===<!-- This section is linked from [[Formal power series]] -->
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| Suppose <math>f(x)=\sum_{n=0}^\infty {a_n} x^n</math>
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| and <math>g(x)=\sum_{n=0}^\infty {b_n} x^n</math>
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| are [[formal power series]] and <math>b_0 = 0</math>.
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| Then the composition <math>f \circ g</math> is again a formal power series,
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| :<math>f(g(x))=\sum_{n=0}^\infty{c_n}x^n,</math>
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| and the coefficient ''c''<sub>''n''</sub>, for ''n ≥ 1'',
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| can be expressed as a sum over compositions of ''n'' or as an equivalent sum over partitions of ''n'' :
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| :<math>c_{n} = \sum_{\mathbf{i}\in \mathcal{C}_{n}} a_{k} b_{i_{1}} b_{i_{2}} \cdots b_{i_{k}}, </math>
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| where
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| :<math>\mathcal{C}_{n}=\{(i_1,i_2,\dots,i_k)\,:\ 1 \le k \le n,\ i_1+i_2+ \cdots + i_k=n\}</math>
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| is the set of compositions of ''n'' with ''k'' denoting the number of parts,
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| or
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| :<math>c_{n} = \sum_{k=1}^{n} a_{k} \sum_{\mathbf{\pi}\in \mathcal{P}_{n,k}} \binom{k}{\pi_{1},\pi_{2}, ..., \pi_{n}} b_{1}^{\pi_{1}} b_{2}^{\pi_{2}}\cdots b_{n}^{\pi_{n}}, </math>
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| where
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| :<math>\mathcal{P}_{n,k}=\{(\pi_1,\pi_2,\dots,\pi_n)\,:\ \pi_1+\pi_2+ \cdots + \pi_n=k,\ \pi_{1}\cdot 1+\pi_{2}\cdot 2+ \cdots + \pi_{n}\cdot n = n \}</math>
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| is the set of partitions of ''n'' into ''k'' parts, in frequency-of-parts form.
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| The first form is obtained by picking out the coefficient of ''x''<sup>''n''</sup>
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| in <math>(b_{1}x+b_{2}x^2+ \cdots)^{k} </math> "by inspection", and the second form
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| is then obtained by collecting like terms, or alternatively, by applying the [[multinomial theorem]].
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| The special case ''f(x) = e<sup>x</sup>, g(x) = ∑<sub>n ≥ 1</sub> a<sub>n </sub>/n! x<sup>n </sup>'' gives the [[exponential formula]].
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| The special case ''f(x) = 1/(1-x), g(x) = ∑<sub>n ≥ 1</sub> (-a<sub>n </sub>) x<sup>n </sup>'' gives an expression for the [[Reciprocal (mathematics)|reciprocal]] of the formal power series '' ∑<sub>n ≥ 0</sub> a<sub>n </sub> x<sup>n </sup>'' in the case ''a<sub>0</sub> = 1'' .
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| Stanley <ref>See the "compositional formula" in Chapter 5 of {{cite book |author=Richard P. Stanley |title=Enumerative Combinatorics |year=1997, 1999 |isbn=0-521-55309-1N |publisher=Cambridge University Press |url=http://www-math.mit.edu/~rstan/ec/}}</ref>
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| gives a version for exponential power series.
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| In the [[formal power series]]
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| :<math>f(x)=\sum_n {a_n \over n!}x^n,</math>
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| we have the ''n''th derivative at 0:
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| :<math>f^{(n)}(0)=a_n. \;</math>
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| 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.
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| If
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| :<math>g(x)=\sum_{n=0}^\infty {b_n \over n!} x^n</math>
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| and
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| :<math>f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n</math>
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| and
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| :<math>g(f(x))=h(x)=\sum_{n=0}^\infty{c_n \over n!}x^n,</math>
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| then the coefficient ''c''<sub>''n''</sub> (which would be the ''n''th derivative of ''h'' evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by
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| :<math>c_n=\sum_{\pi=\left\{\,B_1,\,\dots,\,B_k\,\right\}} a_{\left|B_1\right|}\cdots a_{\left|B_k\right|} b_k</math>
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| where π runs through the set of all partitions of the set { 1, ..., ''n'' } and ''B''<sub>1</sub>, ..., ''B''<sub>''k''</sub> are the blocks of the partition π, and | ''B''<sub>''j''</sub> | is the number of members of the ''j''th block, for ''j'' = 1, ..., ''k''.
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| This version of the formula is particularly well suited to the purposes of [[combinatorics]].
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| We can also write
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| :<math>g(f(x)) = \sum_{n=1}^\infty
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| {\sum_{k=1}^{n} b_k B_{n,k}(a_1,\dots,a_{n-k+1}) \over n!} x^n,</math>
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| where ''B''<sub>''n'',''k''</sub>(''a''<sub>1</sub>,...,''a''<sub>''n''−''k''+1</sub>) are [[Bell polynomials]].
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| ===A special case===
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| If ''f''(''x'') = e<sup>''x''</sup> 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 [[moment (mathematics)|moment]]s as functions of the [[cumulant]]s.
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{Citation
| |
| | first = L. F. A.
| |
| | last=Arbogast
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| | author-link= Louis François Antoine Arbogast
| |
| | title=Du calcul des derivations
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| | year=1800
| |
| | language=[[French language|French]]
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| | publisher=Levrault
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| | place=Strasbourg
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| | url=http://books.google.com/books?id=YoPq8uCy5Y8C
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| | pages= xxiii+404
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| }}, Entirely freely available from [[Google books]].
| |
| *{{Citation
| |
| | 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
| |
| }}.
| |
| *{{Citation
| |
| | first=F.
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| | last=Faà di Bruno
| |
| | author-link=Francesco Faà di Bruno
| |
| | title=Sullo sviluppo delle Funzioni
| |
| | language = [[Italian language|Italian]]
| |
| | journal= Annali di Scienze Matematiche e Fisiche
| |
| | volume=6
| |
| | year=1855
| |
| | pages=479–480
| |
| | url = http://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]].
| |
| *{{Citation
| |
| | first=F.
| |
| | last=Faà di Bruno
| |
| | title= Note sur une nouvelle formule de calcul differentiel
| |
| | language = [[French language|French]]
| |
| | journal= [[The Quarterly Journal of Pure and Applied Mathematics]]
| |
| | volume= 1
| |
| | year=1857
| |
| | pages= 359–360
| |
| | url = http://books.google.com/books?id=7BELAAAAYAAJ&pg=PA359
| |
| }}. Entirely freely available from [[Google books]].
| |
| *{{Citation
| |
| | first = Francesco
| |
| | last= Faà di Bruno
| |
| | title=Théorie générale de l'élimination
| |
| | year=1859
| |
| | language= [[French language|French]]
| |
| | publisher=Leiber et Faraguet
| |
| | place=Paris
| |
| | url=http://books.google.it/books?id=MZ0KAAAAYAAJ
| |
| | pages= x+224
| |
| }}. Entirely freely available from [[Google books]].
| |
| *{{Citation
| |
| | 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
| |
| }}
| |
| *{{Citation
| |
| | 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 = http://www.maa.org/news/monthly217-234.pdf
| |
| | jstor = 2695352
| |
| | doi = 10.2307/2695352
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| | mr = 1903577
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| | zbl = 1024.01010
| |
| }}.
| |
| *{{Citation
| |
| | last = Krantz
| |
| | first = Steven G.
| |
| | author-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 = http://books.google.com/books?id=i4vw2STJl2QC
| |
| | mr = 1916029
| |
| | zbl = 1015.26030
| |
| | isbn = 0-8176-4264-1
| |
| }}
| |
| *{{Citation
| |
| | 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 = http://books.google.com/books?id=BNrW0UJ_UFcC&pg=PA83
| |
| | pages = 83–85
| |
| | pages = xvi+333
| |
| | url = http://books.google.com/books?id=BNrW0UJ_UFcC
| |
| | mr = 1871900
| |
| | zbl = 1013.53001
| |
| | isbn = 0-521-00264-8
| |
| }}.
| |
| *{{Citation
| |
| | last = T. A.
| |
| | first = (Tiburce Abadie, J. F. C.)
| |
| | title = Sur la différentiation des fonctions de fonctions
| |
| | language = [[French language|French]]
| |
| | journal = [http://www.numdam.org/numdam-bin/feuilleter?j=NAM&sl=0 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 [http://www.numdam.org 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.
| |
| *{{Citation
| |
| | last = A.
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| | first = (Tiburce Abadie, J. F. C.)
| |
| | title = Sur la différentiation des fonctions de fonctions. Séries de Burmann, de Lagrange, de Wronski
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| | language = [[French language|French]]
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| | journal = [http://www.numdam.org/numdam-bin/feuilleter?j=NAM&sl=0 Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale]
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| | series = Série 1,
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| | volume = 11
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| | pages = 376–383
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| | year = 1852
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| | url = http://www.numdam.org/item?id=NAM_1852_1_11__376_1
| |
| }}, available at [http://www.numdam.org 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.
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| ==External links==
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| * {{MathWorld|urlname=FaadiBrunosFormula|title=Faa di Bruno's Formula}}
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| * [http://dida.sns.it/dida2/cl/10-11/folde0/pdf20 An intuitive presentation of Faà di Bruno's formula, with examples]
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| {{DEFAULTSORT:Faa de Bruno's formula}}
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| [[Category:Differentiation rules]]
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| [[Category:Differential calculus]]
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| [[Category:Differential algebra]]
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| [[Category:Enumerative combinatorics]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Theorems in analysis]]
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