Faà di Bruno's formula

The formula of Faa di Bruno is a formula of analysis, which was published by the Italian mathematician Francesco Faa di Bruno ( 1825-1888 ).

It can be used higher derivatives of composed functions determine, therefore it generalizes the chain rule and is one of the derivation rules of the differential calculus.

Formulation

Are and twice differentiable functions that depend on a variable and whose composition is well-defined, and is the differential operator according to these variables, the following applies

The amount is summed over here, includes with all tuples of nonnegative integers. That is, the sum extends over all partitions of. The number of summands is therefore the number -th partition. The quotient of the faculties is a multinomial coefficient.

Analogy to the rule of Leibniz

Just as the rule of Leibniz product rule generalized to higher derivatives, so generalizes the formula of Faa di Bruno the chain rule to higher derivatives. The latter formula is, however, evidence and computationally much more difficult.

At the Leibniz rule, there is only summands, whereas significantly more summands occur in the Faa di Bruno 's formula with the- th partition number.

Look at smaller derivation order

If we write the formula for the first natural numbers (or used chain and product rule iteratively ), one sees that the terms are quickly long and unwieldy, and the coefficients are not obvious:

Other derivatives can be calculate with computer algebra systems such as Mathematica or Maple.

Example of use

Using the formula, the coefficients in the Laurent expansion of the gamma function symbolically specify 0. With the functional equation and follows

Where after Faà di Bruno for the -th derivative of the gamma function at the point

Being summed as above on the corresponding set of tuples. The last equal sign the derivatives of the digamma function are used, the Euler - Mascheroni constant and the Riemann zeta function called.