Further Examples: NonCommutativeMultiply (**)

Multiplication is a commutative operation.

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But the noncommutative operator ** does not have this property.

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You can use NonCommutativeMultiply to represent composition in an algebra of differential operators. DOperator[L, expr] applies the differential operator L, written as a pure function, to the expression expr.

The base case, where A is a function, simply multiplies by A.

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The next two properties express linearity.

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Here the operator is D. HoldPattern stops the derivative from acting on the double blank.

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Here a composition of operators is applied to expr.

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Finally a power of an operator is applied to expr.

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The commutator of L1 and L2 measures how close L1 and L2 are to commuting.

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Here is a simple example using these rules.

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Here is another example.

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