New formula for ln(eAeB) in terms of commutators of A and B

We establish the formula $$ln (e^B e^A ) = smallint _0^t psi (e^{ - tau ad_A } e^{ - tau ad_B } ) e^{ - tau ad_A } dtau (A + B),$$ where Ψ(x)=(In x)/(x ? 1); here A and B are elements of a. finite-dimensional Lie algebra which satisfy certain conditions. This formula enables us, in particular, to give a simple proof of the Campbell-Hausdorff theorem. We also give a generalization of the formula to the case of an arbitrary number of factors.