Editing Two-sided Laplace transform (section)

==Region of convergence==
Bilateral transform requirements for convergence are more difficult than for unilateral transforms. The region of convergence will be normally smaller.

If ''f'' is a [[locally integrable]] function (or more generally a [[Borel measure]] locally of [[bounded variation]]), then the Laplace transform ''F''(''s'') of ''f'' converges provided that the limit
: <math>\lim_{R\to\infty}\int_0^R f(t)e^{-st}\, dt</math>
exists. The Laplace transform converges absolutely if the integral
: <math>\int_0^\infty \left|f(t)e^{-st}\right|\, dt</math>
exists (as a proper [[Lebesgue integral]]). The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former instead of the latter sense.

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

Similarly, the set of values for which ''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 ''s'' = ''s''<sub>0</sub>, then it automatically converges for all ''s'' with Re(''s'') > Re(''s''<sub>0</sub>). Therefore, the region of convergence is a half-plane of the form Re(''s'') > ''a'', possibly including some points of the boundary line Re(''s'') = ''a''. In the region of convergence Re(''s'') > Re(''s''<sub>0</sub>), the Laplace transform of ''f'' can be expressed by [[integration by parts|integrating by parts]] as the integral
:<math>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, in the region of convergence ''F''(''s'') can effectively be expressed 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 ''f'' and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to a [[LTI system|linear time-invariant (LTI) system]] is ''stable'' if every bounded input produces a bounded output.