Editing Infinitesimal transformation (section)

==Operator version of Taylor's theorem==
The operator equation

:<math>e^{tD}f(x)=f(x+t)\,</math>

where 

:<math>D={d\over dx}</math>

is an [[Operator (mathematics)|operator]] version of [[Taylor's theorem]] &mdash; and is therefore only valid under ''caveats'' about ''f'' being an [[analytic function]]. Concentrating on the operator part, it shows that ''D'' is an infinitesimal transformation, generating translations of the real line via the [[exponential function|exponential]]. In Lie's theory, this is generalised a long way. Any [[connected space|connected]] Lie group can be built up by means of its [[Lie group#The Lie algebra associated with a Lie group|infinitesimal generator]]s (a basis for the Lie algebra of the group); with explicit if not always useful information given in the [[Baker–Campbell–Hausdorff formula]].