Editing Laplace transform (section)

== Region of convergence ==
{{See also|Pole–zero plot#Continuous-time systems}}
If {{math|''f''}} is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform {{math|''F''(''s'')}} of {{math|''f''}} converges provided that the limit
<math display=block>\lim_{R\to\infty}\int_0^R f(t)e^{-st}\,dt</math>
exists.

The Laplace transform [[Absolute convergence|converges absolutely]] if the integral
<math display=block>\int_0^\infty \left|f(t)e^{-st}\right|\,dt</math>
exists as a proper Lebesgue integral. The Laplace transform is usually understood as [[Conditional convergence|conditionally convergent]], meaning that it converges in the former but not in the latter sense.

The set of values for which {{math|''F''(''s'')}} converges absolutely is either of the form {{math|Re(''s'') > ''a''}} or {{math|Re(''s'') ≥ ''a''}}, where {{math|''a''}} is an [[extended real number|extended real constant]] with {{math|−∞ ≤ ''a'' ≤ ∞}} (a consequence of the [[dominated convergence theorem]]). The constant {{math|''a''}} is known as the abscissa of absolute convergence, and depends on the growth behavior of {{math|''f''(''t'')}}.<ref>{{harvnb|Widder|1941|loc=Chapter II, §1}}</ref> Analogously, the two-sided transform converges absolutely in a strip of the form  {{math|''a'' < Re(''s'') < ''b''}}, and possibly including the lines {{math|1=Re(''s'') = ''a''}} or {{math|1=Re(''s'') = ''b''}}.<ref>{{harvnb|Widder|1941|loc=Chapter VI, §2}}</ref> The subset of values of {{math|''s''}} for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence.  In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of [[Fubini's theorem]] and [[Morera's theorem]].

Similarly, the set of values for which {{math|''F''(''s'')}} converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the '''region of convergence''' (ROC). If the Laplace transform converges (conditionally) at {{math|1=''s'' = ''s''<sub>0</sub>}}, then it automatically converges for all {{math|''s''}} with {{math|Re(''s'') > Re(''s''<sub>0</sub>)}}. Therefore, the region of convergence is a half-plane of the form {{math|Re(''s'') > ''a''}}, possibly including some points of the boundary line {{math|1=Re(''s'') = ''a''}}.

In the region of convergence {{math|Re(''s'') > Re(''s''<sub>0</sub>)}}, the Laplace transform of {{math|''f''}} can be expressed by [[integration by parts|integrating by parts]] as the integral
<math display=block>F(s) = (s-s_0)\int_0^\infty e^{-(s-s_0)t}\beta(t)\,dt, \quad \beta(u) = \int_0^u e^{-s_0t}f(t)\,dt.</math>

That is, {{math|''F''(''s'')}} can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function.  In particular, it is analytic.

There are several [[Paley–Wiener theorem]]s concerning the relationship between the decay properties of {{math|''f''}}, and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to a [[Linear time-invariant system|linear time-invariant (LTI) system]] is ''stable'' if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region {{math|Re(''s'') ≥ 0}}. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.

This ROC is used in knowing about the causality and stability of a system.