= Commutator of an operator with an exponential
{title2=$[A,e^{-\beta B}]$}
For bounded <linear operators>, define $C(\beta)=[A,e^{-\beta B}]$. The <product rule> gives $C'+BC=[B,A]e^{-\beta B}$ and $C(0)=0$. Multiplying by the <integrating factor> $e^{\beta B}$ yields
$$
[A,e^{-\beta B}]
=e^{-\beta B}\int_0^\beta e^{\lambda B}[B,A]e^{-\lambda B}\,d\lambda.
$$
The identity is useful when the two <linear operators> do not commute: it preserves their order rather than incorrectly differentiating an <operator exponential> as if all its factors were scalars. For unbounded <linear operators>, the products and differentiations need a common justified <operator domain>.
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