Ordered exponential

Ordered Exponential is the mathematical object, defined in the non-commutative algebras, which is equivalent to the normal exponential function of the integral in the commutative algebras. In practice such objects are observed in the matrix and operator algebras.

For the element A(t) from the algebra (set g with the non-commutative product *), where t is the time parameter, the ordered exponential of A can be defined via one of several equivalent approaches:

• As the limit of the ordered product of the infinitesimal exponentials:

where the time moments are defined as for , and .

• Via the Initial Value Problem, where the OE[A](t) is the unique solution of the system of equations:

• Via following integral equation:

• Via Taylor series expansion: