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The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin. (English) Zbl 1245.01002
“One of the most versatile results originating from the early theory of groups of transformations $$(\dots)$$ states that, in the algebra of formal series in two non-commuting indeterminates $$x$$ and $$y$$, the series naturally associated to $$\log(e^xe^y)$$ is a series of Lie polynomials in $$x$$ and $$y$$.” The result, sometimes called the Exponential Theorem, has found important applications to physics, group theory, linear PDEs, Lie groups and Lie algebras, numerical analysis. The aim of this paper is to recall nearly forgotten contributions given by the forerunners of the Theorem, in particular by Italian mathematician E. Pascal, whose work had been “of decisive importance”. With the intention to ease the access to those contributions the authors furnish the mathematical details and offer an explanation in modern language.
The paper is divided into sections. After an introduction there follow sections on contributions by F. Schur, J. E. Campbell, H. Poincaré, E. Pascal, H. F. Baker, F. Hausdorff, and E. B. Dynkin, with the final one on commentaries by F. Hausdorff and N. Bourbaki which, in the authors’ opinion, were rather “cold” and thus played a major role in subsequent neglecting those early contributions which still seem to be of some value. – The paper is completed with an extensive bibliography.

##### MSC:
 01-02 Research exposition (monographs, survey articles) pertaining to history and biography 01A55 History of mathematics in the 19th century 01A60 History of mathematics in the 20th century 22-03 History of topological groups
##### Keywords:
Pascal; Lie-groups; Hausdorff; Bourbaki
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