Faà di Bruno's formula について

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Faà di Bruno's formula ：ウィキペディア英語版

Faà di Bruno's formula

Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after , 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, considered the first published reference on the subject.
Perhaps the most well-known form of Faà di Bruno's formula says that
:

f(g(x))=\sum \frac\,\cdots\,m_n!\,n!^}\cdot f^(g(x))\cdot \prod_^n\left(g^(x)\right)^,

where the sum is over all ''n''-tuples of nonnegative integers (''m''₁, …, ''m''_''n'') satisfying the constraint
:

1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+n\cdot m_n=n.\,

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:
:

f(g(x))=\sum \frac\cdotf^(g(x))\cdot\prod_^n\left(\frac\right)^.

Combining the terms with the same value of ''m''₁ + ''m''₂ + ... + ''m''_''n'' = ''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 ''B''_''n'',''k''(''x''₁,...,''x''_{''n''−''k''+1}):
:

f(g(x)) = \sum_^n f^(g(x))\cdot B_\left(g'(x),g''(x),\dots,g^(x)\right).

== Combinatorial form ==

The formula has a "combinatorial" form:
:

f(g(x))=(f\circ g)^(x)=\sum_ f^(g(x))\cdot\prod_g^(x)

where
*π runs through the set Π of all partitions of the set ,
*"''B'' ∈ π" means the variable ''B'' runs through the list of all of the "blocks" of the partition π, and
*|''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'').

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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