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Baker-Campbell-Hausdorff Series

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The series

z=ln(e^xe^y)
(1)

for noncommuting variables x and y. The first few terms are

z_1=x+y
(2)
z_2=1/2(xy-yx)
(3)
z_3=1/(12)(x^2y+xy^2-2xyx+y^2x+yx^2-2yxy)
(4)
z_4=1/(24)(x^2y^2-2xyxy-y^2x^2+2yxyx).
(5)

The series can also be generalized to

z=ln(e^xe^ye^w)
(6)

(Reinsch 2000), giving the first few terms as

z_1=x+y+w
(7)
z_2=1/2(-wx-wy+xw+xy+yw-yx).
(8)

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References

Bose, A. "Dynkin's Method of Computing the Terms of the Baker-Campbell-Hausdorff Series." J. Math. Phys. 30, 2035-2037, 1989.Dynkin, E. B. "On the Representation by Means of Commutators of the Series log(e^xe^y) for Noncommuting x,y." Mat. Sb. 25, 155-162, 1949.Gilmore, R. "Baker-Campbell-Hausdorff Formulas." J. Math. Phys. 15, 2090-2092, 1974.Goldberg, K. "The Formal Power Series for loge^xe^y." Duke Math. J. 23, 13-21, 1956.Kobayashi, H.; Hatano, N.; and Suzuki, M. "Goldberg's Theorem and the Baker-Campbell-Hausdorff Formula." Physica A 250, 535-548, 1998.Magnus, W. A. Ann. Math. 52, 111, 1952.Magnus, W. A. Comm. Pure Appl. Math. 7, 649, 1954.Munthe-Kaas, H. and Owren, B. "Computations in a Free Lie Algebra." Philos. Trans. Roy. Soc. London A 357, 957-981, 1999.Newman, M. and Thompson, R. C. "Numerical Values of Goldberg's Coefficients in the Series for log(e^xe^y)." Math. Comput. 48, 265-271, 1987.Oteo, J. A. "The Baker-Campbell-Hausdorff Formula and Nested Commutator Identities." J. Math. Phys. 32, 419-424, 1991.Reinsch, M. W. "A Simple Expression for the Terms in the Baker-Campbell-Hausdorff Series." 13 Jan 2000. http://arxiv.org/abs/math-ph/9905012.Reutenauer, C. "Algebraic Properties." Ch. 3 in Free Lie Algebras. New York: Oxford University Press, 1993.Thompson, R. C. "Cyclic Relations and the Goldberg Coefficients in the Campbell-Baker-Hausdorff Formula." Proc. Amer. Math. Soc. 86, 12-14, 1982.Weiss, G. H. and Maradudin, A. A. "The Baker-Hausdorff Formula and a Problem in Crystal Physics." J. Math. Phys. 3, 771-777, 1962.

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Baker-Campbell-Hausdorff Series

Cite this as:

Weisstein, Eric W. "Baker-Campbell-Hausdorff Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Baker-Campbell-HausdorffSeries.html

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