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(view preamble)
| Other names: |
BCH formula, Baker-Campbell-Hausdorff formula, Baker-Campbell-Hausdorff formulae |
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Cross-references: factor, exponentials, multiplication, expression, representation, series, linear operator
There are 2 references to this entry.
This is version 10 of Baker-Campbell-Hausdorff formula(e), born on 2003-06-04, modified 2008-02-11.
Object id is 4321, canonical name is BakerCampellHausdorffFormulae.
Accessed 26243 times total.
Classification:
| AMS MSC: | 47A05 (Operator theory :: General theory of linear operators :: General ) |
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Pending Errata and Addenda
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![$ B_m =[A,B]_m := [A,[A,B]_{m-1}]$](http://images.planetmath.org:8080/cache/objects/4321/l2h/img8.png "$ B_m =[A,B]_m := [A,[A,B]_{m-1}]$"){kind=small}
![$\displaystyle [B,e^{-A}] = e^{-A} \left( [A,B]+\frac{1}{2} [A,[A,B]] + \cdots \right),$](http://images.planetmath.org:8080/cache/objects/4321/l2h/img12.png "$\displaystyle [B,e^{-A}] = e^{-A} \left( [A,B]+\frac{1}{2} [A,[A,B]] + \cdots \right),$")
![$\displaystyle [e^A,B] = \left( [A,B]+\frac{1}{2} [A,[A,B]] + \cdots \right) e^A.$](http://images.planetmath.org:8080/cache/objects/4321/l2h/img13.png "$\displaystyle [e^A,B] = \left( [A,B]+\frac{1}{2} [A,[A,B]] + \cdots \right) e^A.$")
![$ [A,[A,B]] = [B,[B,A]] = 0$](http://images.planetmath.org:8080/cache/objects/4321/l2h/img14.png "$ [A,[A,B]] = [B,[B,A]] = 0$"){kind=small}
![$\displaystyle e^A e^B = e^{A+B} e^{\frac{1}{2}[A,B]}.$](http://images.planetmath.org:8080/cache/objects/4321/l2h/img15.png "$\displaystyle e^A e^B = e^{A+B} e^{\frac{1}{2}[A,B]}.$"){kind=small}
![$\displaystyle e^A e^B = e^B e^A e^{[A,B]}.$](http://images.planetmath.org:8080/cache/objects/4321/l2h/img16.png "$\displaystyle e^A e^B = e^B e^A e^{[A,B]}.$"){kind=small}