Formula by Faà di Bruno

The formula of Faà di Bruno is a formula of analysis published by the Italian mathematician Francesco Faà di Bruno (1825–1888).

It can be used to determine higher derivatives of composed functions , it thus generalizes the chain rule and belongs to the derivation rules of differential calculus .

formulation

If and are two- times differentiable functions, which depend on a variable and whose composition is well-defined, and if the differential operator is after this variable, then we have ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle D}$

${\displaystyle D^{n}(f\circ g)=\sum _{(k_{1},\,\ldots \,,k_{n})\in T_{n}}{\frac {n!}{k_{1}!\cdot \ \cdots \ \cdot k_{n}!}}{\bigl (}D^{k_{1}+\ldots +k_{n}}f\circ g{\bigr )}\,\prod _{m=1 \atop k_{m}\geq 1}^{n}{\biggl (}{\frac {D^{m}g}{m!}}{\biggr )}^{k_{m}}\,}$.

The set that is summed over here contains all - tuples of non-negative , integers with . Each such tuple can be mapped bijectively on a partition of in which the summand Mal occurs. The number of summands is therefore the -th partition number . The quotient of the faculties is a multinomial coefficient . ${\displaystyle T_{n}}$ ${\displaystyle n}$ ${\displaystyle (k_{1},\ \ldots \ ,k_{n})\,}$ ${\displaystyle 1k_{1}+2k_{2}+\cdots +nk_{n}=n\,}$ ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle k_{i}}$ ${\displaystyle n}$

Analogy to Leibniz's rule

Just as Leibniz 's rule generalizes the product rule to higher derivatives, so Faà di Bruno's formula generalizes the chain rule to higher derivatives. However, the latter formula is much more difficult in terms of evidence and computation.

In Leibniz's rule there are only summands, whereas in Faà di Bruno's formula with the -th partition number there are significantly more summands. ${\displaystyle n+1}$ ${\displaystyle n}$ ${\displaystyle P(n)}$

Appearance with a small order of derivation

If you write out the formula for the first natural numbers (or use chain and product rule iteratively), you can see that the expressions quickly become long and unwieldy and the coefficients are not obvious:

${\displaystyle {\begin{aligned}D(f\circ g)&={\bigl (}f'\circ g{\bigr )}\,g'\\D^{2}(f\circ g)&={\bigl (}f''\circ g{\bigr )}\,(g')^{2}+{\bigl (}f'\circ g{\bigr )}\,g''\\D^{3}(f\circ g)&={\bigl (}f'''\circ g{\bigr )}\,(g')^{3}+3\,{\bigl (}f''\circ g{\bigr )}\,g'\,g''+{\bigl (}f'\circ g{\bigr )}\,g'''\\D^{4}(f\circ g)&={\bigl (}f''''\circ g{\bigr )}\,(g')^{4}+6\,{\bigl (}f'''\circ g{\bigr )}\,(g')^{2}\,g''\\&\quad +4\,{\bigl (}f''\circ g{\bigr )}\,g'\,g'''+3\,{\bigl (}f''\circ g{\bigr )}\,(g'')^{2}+{\bigl (}f'\circ g{\bigr )}\,g''''\\D^{5}(f\circ g)&={\bigl (}f'''''\circ g{\bigr )}\,(g')^{5}+10\,{\bigl (}f''''\circ g{\bigr )}\,(g')^{3}\,g''\\&\quad +10\,{\bigl (}f'''\circ g{\bigr )}\,(g')^{2}\,g'''+15\,{\bigl (}f'''\circ g{\bigr )}\,g'\,(g'')^{2}\\&\quad +10\,{\bigl (}f''\circ g{\bigr )}\,g''\,g'''+5\,{\bigl (}f''\circ g{\bigr )}\,g'\,g''''+{\bigl (}f'\circ g{\bigr )}\,g'''''\end{aligned}}}$

Further derivations can be calculated with computer algebra systems such as Mathematica or Maple .

Application when concatenating power series

Are and two power series ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-x_{1})^{n}}$

${\displaystyle g(x)=\sum _{n=0}^{\infty }b_{n}(x-x_{0})^{n}}$

with positive radii of convergence and the property

${\displaystyle g(x_{0})=x_{1}}$

Then the concatenation of both functions is locally again an analytic function and can therefore be developed into a power series: ${\displaystyle f\circ g}$ ${\displaystyle x_{0}}$

${\displaystyle (f\circ g)(x)=\sum _{n=0}^{\infty }c_{n}(x-x_{0})^{n}}$

According to Taylor's theorem:

${\displaystyle c_{n}={\frac {(f\circ g)^{(n)}(x_{0})}{n!}}}$

With Faà di Bruno's formula, this expression can now be given in a closed formula depending on the given series coefficients, since:

${\displaystyle {\begin{aligned}f^{(n)}(g(x_{0}))&=f^{(n)}(x_{1})\\&=n!\cdot a_{n}\\g^{(m)}(x_{0})&=m!\cdot b_{m}\end{aligned}}}$

With multi-index notation you get :

${\displaystyle {\begin{aligned}c_{n}&={\frac {(f\circ g)^{(n)}(x_{0})}{n!}}\\&=\sum _{{\boldsymbol {k}}\in T_{n}}{\frac {f^{(|{\boldsymbol {k}}|)}(g(x_{0}))}{{\boldsymbol {k}}!}}\prod _{m=1 \atop k_{m}\geq 1}^{n}\left({\frac {g^{(m)}(x_{0})}{m!}}\right)^{k_{m}}\\&=\sum _{{\boldsymbol {k}}\in T_{n}}{\frac {|{\boldsymbol {k}}|!\cdot a_{|{\boldsymbol {k}}|}}{{\boldsymbol {k}}!}}\prod _{m=1 \atop k_{m}\geq 1}^{n}\left(b_{m}\right)^{k_{m}}\\&=\sum _{{\boldsymbol {k}}\in T_{n}}{{|{\boldsymbol {k}}|} \choose {\boldsymbol {k}}}\,a_{|{\boldsymbol {k}}|}\prod _{m=1 \atop k_{m}\geq 1}^{n}b_{m}^{k_{m}}\end{aligned}}}$

It is the multinomial to and is again the set of partition (see partition function ). ${\displaystyle {{|{\boldsymbol {k}}|} \choose {\boldsymbol {k}}}}$ ${\displaystyle {\boldsymbol {k}}}$ ${\displaystyle T_{n}=\left\{{\boldsymbol {k}}\in \mathbb {N} _{0}^{n}\,{\Big |}\,\sum _{j=1}^{n}j\cdot k_{j}=n\right\}}$ ${\displaystyle n}$

Application example

With the help of the formula, the coefficients in the Laurent series of the gamma function can be specified symbolically in 0. With the functional equation and follows ${\displaystyle \Gamma (1)=1}$

${\displaystyle \Gamma (x)={\frac {\Gamma (1+x)}{x}}={\frac {1}{x}}\sum _{n=0}^{\infty }{\frac {D^{n}\Gamma (1)}{n!}}x^{n}={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {D^{n}\Gamma (1)}{n!}}x^{n-1}}$.

According to Faà di Bruno, the -th derivative of the gamma function at the point applies ${\displaystyle n}$ ${\displaystyle 1}$

${\displaystyle {\begin{aligned}D^{n}\Gamma (1)&=D^{n}e^{\ln \Gamma (1)}\\&=\sum _{(k_{1},\dots ,k_{n})\in T_{n}}{\frac {n!}{k_{1}!\cdots k_{n}!}}\,\Gamma (1)\prod _{m=1 \atop k_{m}\geq 1}^{n}\left({\frac {D^{m}\ln \Gamma (1)}{m!}}\right)^{k_{m}}\\&=\sum _{(k_{1},\dots ,k_{n})\in T_{n}}{\frac {n!}{k_{1}!\cdots k_{n}!}}\,(-\gamma )^{k_{1}}\prod _{m=2 \atop k_{m}\geq 1}^{n}\left((-1)^{m}\,{\frac {\zeta (m)}{m}}\right)^{k_{m}},\end{aligned}}}$

where, as above, the corresponding set of tuples is added up. For the last equal sign, the derivatives of the digamma function are used, where the Euler-Mascheroni constant and the Riemann zeta function are used. ${\displaystyle T_{n}}$ ${\displaystyle n}$ ${\displaystyle \psi (z)={\tfrac {\Gamma '(z)}{\Gamma (z)}}}$ ${\displaystyle \gamma =-\psi (1)}$ ${\displaystyle \zeta }$

Web links

• Partition numbers are the sequence A000041 in OEIS
• Monthly (PDF file; 193 kB)
• Mathworld