Faà di Bruno's formula is an identity in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
generalizing the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
to higher derivatives. It is named after , although 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 had stated the formula in a calculus textbook, which is considered to be the first published reference on the subject.
Perhaps the most well-known form of Faà di Bruno's formula says that
where the sum is over all
-
tuple
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...
s of nonnegative integers
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
and noticing that
has to be zero for
leads to a somewhat simpler formula expressed in terms of partial (or incomplete) exponential
Bell polynomials
:
:
This formula works for all
, however for
the polynomials
are zero and thus summation in the formula can start with
.
Combinatorial form
The formula has a "combinatorial" form:
:
where
*
runs through the set
of all
partitions of the set ,
*"
" means the variable
runs through the list of all of the "blocks" of the partition
, and
*
denotes the cardinality of the set
(so that
is the number of blocks in the partition
and
is the size of the block
).
Example
The following is a concrete explanation of the combinatorial form for the
case.
:
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:
:
Variations
Multivariate version
Let
. Then the following identity holds regardless of whether the
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):
:
where (as above)
*
runs through the set
of all
partitions of the set ,
*"
" means the variable
runs through the list of all of the "blocks" of the partition
, and
*
denotes the cardinality of the set
(so that
is the number of blocks in the partition
and
is the size of the block
).
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.
; Example
The five terms in the following expression correspond in the obvious way to the five partitions of the set
, and in each case the order of the derivative of
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
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
and
.
Then the composition
is again a formal
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
,
:
where
and the other coefficient
for
can be expressed as a sum over
compositions of
or as an equivalent sum over
integer partitions of
:
:
where
:
is the set of compositions of
with
denoting the number of parts,
or
:
where
:
is the set of partitions of
into
parts, in frequency-of-parts form.
The first form is obtained by picking out the coefficient of
in
"by inspection", and the second form
is then obtained by collecting like terms, or alternatively, by applying the
multinomial theorem
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.
Theorem
For any positive integer ...
.
The special case
,
gives the
exponential formula.
The special case
,
gives an expression for the
reciprocal of the formal power series
in the case
.
Stanley
[See the "compositional formula" in Chapter 5 of ]
gives a version for exponential power series.
In the
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
:
we have the
th 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
(which would be the
th derivative of
evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by
:
where
runs through the set of all partitions of the set
and
are the blocks of the partition
, and
is the number of members of the
th block, for
.
This version of the formula is particularly well suited to the purposes of
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
.
We can also write with respect to the notation above
:
where
are
Bell polynomials.
A special case
If
, then all of the derivatives of
are the same and are a factor common to every term:
:
where
is the ''n''th
complete exponential Bell polynomial.
In case
is a
cumulant-generating function, then
is a
moment-generating function, and the polynomial in various derivatives of
is the polynomial that expresses the
moments as functions of the
cumulants.
See also
*
*
*
*
*
*
*
*
*
Notes
References
Historical surveys and essays
*. "''The mathematical work''" is an essay on the mathematical activity, describing both the research and teaching activity of Francesco Faà di Bruno.
*.
*.
Research works
*, Entirely freely available from
Google books
Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google that searches the full text of books and magazines that Google has scanned, converted to text using optical charac ...
.
*. Entirely freely available from
Google books
Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google that searches the full text of books and magazines that Google has scanned, converted to text using optical charac ...
. 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.
*. Entirely freely available from
Google books
Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google that searches the full text of books and magazines that Google has scanned, converted to text using optical charac ...
.
*. Entirely freely available from
Google books
Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google that searches the full text of books and magazines that Google has scanned, converted to text using optical charac ...
.
*
Flanders, Harley (2001) "From Ford to Faa",
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
108(6): 558–61
*.
*
*.
*, available a
NUMDAM This paper, according to is one of the precursors of : note that the author signs only as "T.A.", and the attribution to J. F. C. Tiburce Abadie is due again to Johnson.
*, available a
NUMDAM This paper, according to is one of the precursors of : note that the author signs only as "A.", and the attribution to J. F. C. Tiburce Abadie is due again to Johnson.
External links
*
{{DEFAULTSORT:Faa de Bruno's formula
Differentiation rules
Differential calculus
Differential algebra
Enumerative combinatorics
Factorial and binomial topics
Theorems in mathematical analysis