Exponential product defect identity
= Exponential product defect identity
{title2=$e^{tA}e^{tB}-e^{t(A+B)}$}
For finite <matrices>, differentiate the ordered product to obtain $\Phi'-(A+B)\Phi=[e^{tA},B]e^{tB}$. Multiplying the error by $e^{-t(A+B)}$ and integrating gives $\Phi(t)-e^{t(A+B)}=\int_0^t e^{(t-s)(A+B)}[e^{sA},B]e^{sB}\,ds$. This exact <variation-of-constants formula> identifies noncommutativity as the source of <Lie-Trotter splitting> error.