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An exact power series representation of the Baker–Campbell–Hausdorff formula
Journal of Physics A: Mathematical and Theoretical ( IF 2.1 ) Pub Date : 2020-12-15 , DOI: 10.1088/1751-8121/abcbae
Jordan C Moodie , M W Long

An exact representation of the Baker–Campbell–Hausdorff formula as a power series in just one of the two variables is constructed. Closed form coefficients of this series are found in terms of hyperbolic functions, which contain all of the dependence on the second variable. It is argued that this exact series may then be truncated and be expected to give a good approximation to the full expansion if only the perturbative variable is small. This improves upon existing formulae, which require both to be small. Several different representations are provided and emphasis is given to the situation where one of the matrices is diagonal, where a particularly easy to use formula is obtained.



中文翻译:

Baker–Campbell–Hausdorff公式的精确幂级数表示

仅在两个变量之一中构造了贝克-坎贝尔-豪斯道夫公式作为幂级数的精确表示。根据双曲函数可以找到该级数的封闭形式系数,该函数包含对第二变量的所有依赖关系。有人认为,如果仅摄动变量较小,则该精确序列可能会被截断,并有望为全展开提供良好的近似值。这改进了要求两者都小的现有公式。提供了几种不同的表示形式,并着重说明了其中一个矩阵是对角线的情况,其中获得了一个特别易于使用的公式。

更新日期:2020-12-15
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