General Leibniz rule

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For other uses, see Leibniz's rule (disambiguation).

In calculus, the general Leibniz rule,[1] named after Gottfried Leibniz, generalizes the product rule (which is also known as "Leibniz's rule".) It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by

(f \cdot g)^{(n)}=\sum_{k=0}^n {n \choose k} f^{(k)} g^{(n-k)}

where {n \choose k} is the binomial coefficient.

This can be proved by using the product rule and mathematical induction.

With the multi-index notation the rule states more generally:

\partial^\alpha (fg) = \sum_{ \{\beta\,:\,\beta \le \alpha \} } {\alpha \choose \beta} (\partial^{\alpha - \beta} f) (\partial^{\beta} g).

This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let P and Q be differential operators (with coefficients that are differentiable sufficiently many times) and R = P \circ Q. Since R is also a differential operator, the symbol of R is given by:

R(x, \xi) = e^{-{\langle x, \xi \rangle}} R (e^{\langle x, \xi \rangle}).

A direct computation now gives:

R(x, \xi) = \sum_\alpha {1 \over \alpha!} \left({\partial \over \partial \xi}\right)^\alpha P(x, \xi) \left({\partial \over \partial x}\right)^\alpha Q(x, \xi).

This formula is usually known as the Leibniz formula. It is used to define the composition in the space of symbols, thereby inducing the ring structure.

See also[edit]


  1. ^ Olver, Applications of Lie groups to differential equations, page 318

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