Differential operators

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).

This article considers mainly linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative also exist.

Assume that there is a map ${\displaystyle A}$ from a function space ${\displaystyle {\mathcal {F}}_{1}}$ to another function space ${\displaystyle {\mathcal {F}}_{2}}$ and a function ${\displaystyle f\in {\mathcal {F}}_{2}}$ so that ${\displaystyle f}$ is the image of ${\displaystyle u\in {\mathcal {F}}_{1}}$ i.e.,　${\displaystyle f=A(u)\ .}$ A differential operator is represented as a linear combination, finitely generated by ${\displaystyle u}$ and its derivatives containing higher degree such as

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