Editing Differential operator (section)

==Properties==

Differentiation is [[linear map|linear]], i.e.

:<math>D(f+g) = (Df)+(Dg),</math>
:<math>D(af) = a(Df),</math>

where ''f'' and ''g'' are functions, and ''a'' is a constant.

Any [[polynomial]] in ''D'' with function coefficients is also a differential operator. We may also [[function composition|compose]] differential operators by the rule

:<math>(D_1 \circ D_2)(f) = D_1(D_2(f)).</math>

Some care is then required: firstly any function coefficients in the operator ''D''<sub>2</sub> must be [[Differentiable function|differentiable]] as many times as the application of ''D''<sub>1</sub> requires. To get a [[ring (mathematics)|ring]] of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be [[commutative ring|commutative]]: an operator ''gD'' isn't the same in general as ''Dg''. For example we have the relation basic in [[quantum mechanics]]:
:<math>Dx - xD = 1.</math>

The subring of operators that are polynomials in ''D'' with [[constant coefficients]] is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.

The differential operators also obey the [[shift theorem]].