Editing Exponential function (section)

==Matrices and Banach algebras==
The power series definition of the exponential function makes sense for square [[matrix (mathematics)|matrices]] (for which the function is called the [[matrix exponential]]) and more generally in any unital [[Banach algebra]] {{math|''B''}}.  In this setting, {{math|1=''e''{{isup|0}} = 1}}, and {{math|''e''{{isup|''x''}}}} is invertible with inverse {{math|''e''{{isup|−''x''}}}} for any {{math|''x''}} in {{math|''B''}}.  If {{math|1=''xy'' = ''yx''}}, then {{math|1=''e''{{isup|''x'' + ''y''}} = ''e''{{isup|''x''}}''e''{{isup|''y''}}}}, but this identity can fail for noncommuting {{math|''x''}} and {{math|''y''}}.

Some alternative definitions lead to the same function.  For instance, {{math|''e''{{isup|''x''}}}} can be defined as
<math display="block">\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n .</math>

Or {{math|''e''{{isup|''x''}}}} can be defined as {{math|''f''<sub>''x''</sub>(1)}}, where {{math|''f''<sub>''x''</sub> : '''R''' → ''B''}} is the solution to the differential equation {{math|1={{sfrac|''df''<sub>''x''</sub>|''dt''}}(''t'') = ''x{{space|hair}}f''<sub>''x''</sub>(''t'')}}, with initial condition {{math|1=''f''<sub>''x''</sub>(0) = 1}}; it follows that {{math|1=''f''<sub>''x''</sub>(''t'') = ''e''{{isup|''tx''}}}} for every {{mvar|t}} in {{math|'''R'''}}.