Editing Ordered exponential (section)

=== Product of exponentials ===

The ordered exponential can be defined as the left [[product integral]] of the [[infinitesimal]] exponentials, or equivalently, as an [[ordered product]] of exponentials in the [[Limit (mathematics)|limit]] as the number of terms grows to infinity:
:<math>\operatorname{OE}[a](t) = \prod_0^t e^{a(t') \, dt'} \equiv
    \lim_{N \to \infty} \left(
      e^{a(t_N) \, \Delta t} e^{a(t_{N-1}) \, \Delta t} \cdots
      e^{a(t_1) \, \Delta t} e^{a(t_0) \, \Delta t}
    \right)
  </math>

where the time moments {{math|{{mset|''t''<sub>0</sub>, ..., ''t''<sub>''N''</sub>}}}} are defined as {{math|''t''<sub>''i''</sub> ≡ ''i'' Δ''t''}} for {{math|1=''i'' = 0, ..., ''N''}}, and {{math|Δ''t'' ≡ ''t'' / ''N''}}.

The ordered exponential is in fact a [[Product integral#Type II|geometric integral]]{{Broken anchor|date=2024-05-27|bot=User:Cewbot/log/20201008/configuration|target_link=Product integral#Type II|reason= The anchor (Type II) [[Special:Diff/835454693|has been deleted]].}}.<ref name=nnc>Michael Grossman and Robert Katz. [https://books.google.com/books?q=%22Non-Newtonian+Calculus%22&btnG=Search+Books&as_brr=0 ''Non-Newtonian Calculus''], {{ISBN|0912938013}}, 1972.</ref><ref>A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. [http://linkinghub.elsevier.com/retrieve/pii/S0022247X07003824 ''Multiplicative calculus and its applications''], Journal of Mathematical Analysis and Applications, 2008.</ref><ref name=FvA>Luc Florack and Hans van Assen.[https://doi.org/10.1007%2Fs10851-011-0275-1 "Multiplicative calculus in biomedical image analysis"], Journal of Mathematical Imaging and Vision, 2011. </ref>