Geodesic
New formula for $\ln(e^Ae^B)$ in terms of commutators of $A$ and $B$
Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 817-824

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We establish the formula $$ \ln(e^Be^A)=\int_0^t\psi(e^{-\tau ad_A}e^{-\tau ad_B})e^{-\tau ad_A}\,d\tau(A+B), $$ where $\psi(x)=(\ln x)/(x-1)$; 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. 
@article{MZM_1978_23_6_a3,
     author = {M. V. Mosolova},
     title = {New formula for $\ln(e^Ae^B)$ in terms of commutators of $A$ and $B$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {817--824},
     publisher = {mathdoc},
     volume = {23},
     number = {6},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_6_a3/}
}
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M. V. Mosolova. New formula for $\ln(e^Ae^B)$ in terms of commutators of $A$ and $B$. Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 817-824. http://geodesic.mathdoc.fr/item/MZM_1978_23_6_a3/