Ursell function: Difference between revisions

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In [[mathematics]], the '''Lie product formula''', named for [[Sophus Lie]], states that for arbitrary ''n'' × ''n'' [[real number|real]] or [[complex number|complex]] [[matrix (mathematics)|matrices]] ''A'' and ''B'',
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:<math>e^{A+B} = \lim_{N \rightarrow \infty} (e^{A/N}e^{B/N})^N, </math>
 
where ''e''<sup>''A''</sup> denotes the [[matrix exponential]] of ''A''.  
The '''Lie–Trotter product formula''' {{harv|Trotter|1959}} and the '''Trotter–Kato theorem''' {{harv|Kato|1978}} extend this to certain unbounded linear operators ''A'' and ''B''.  
 
This formula is an analogue of the classical exponential law
 
:<math>e^{x+y} = e^{x}e^{y} \,</math>
 
which holds for all real or complex numbers ''x'' and ''y''. If ''x'' and ''y'' are replaced with matrices ''A'' and ''B'', and the [[Exponential function|exponential]] replaced with a [[matrix exponential]], it is usually necessary for ''A'' and ''B'' to commute for the law to still hold. However, the Lie product formula holds for all matrices ''A'' and ''B'', even ones which do not commute.  
 
It is a trivial corollary of the [[Baker–Campbell–Hausdorff formula]].
 
The formula has applications, for example, in the [[path integral formulation]] of quantum mechanics. It allows one to separate the [[Propagator|Schrödinger evolution operator]] into alternating increments of kinetic and potential operators. The same idea is used in the construction of [[Symplectic integrator|splitting method]]s for the numerical solution of [[differential equation]]s.
 
== References ==
* {{Citation | last1=Albeverio | first1=Sergio A. | last2=Høegh-Krohn | first2=Raphael J. | title=Mathematical Theory of Feynman Path Integrals: An Introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=1st | series=Lecture Notes in Mathematics | isbn=978-3-540-07785-5 | doi=10.1007/BFb0079827 | year=1976 | volume=423}}.
* {{Citation | first=Brian C. | last=Hall | title=Lie groups, Lie algebras, and representations: an elementary introduction | publisher=Springer | year=2003 | isbn=978-0-387-40122-5}}, pp. 35.
*{{eom|id=T/t094340|title=Trotter product formula}}
*{{Citation | last1=Kato | first1=Tosio | title=Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday) | publisher=[[Academic Press]] | location=Boston, MA | series=Adv. in Math. Suppl. Stud. | id={{MathSciNet | id = 538020}} | year=1978 | volume=3 | chapter=Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups | pages=185–195}}
*{{Citation | last1=Trotter | first1=H. F. | title=On the product of semi-groups of operators | id={{MathSciNet | id = 0108732}} | year=1959 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=10 | issue=4 | pages=545–551 | doi=10.2307/2033649 | jstor=2033649}}
* {{Citation | first=V.S. | last=Varadarajan | title=Lie Groups, Lie Algebras, and Their Representations | publisher=Springer-Verlag | year=1984 | isbn=978-0-387-90969-1}}, pp. 99.
 
[[Category:Matrix theory]]
[[Category:Lie groups]]

Latest revision as of 01:20, 26 September 2014

How To Make Defense Runes In "League Of Legends"


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