Editing Laplace transform (section)

=== Fourier transform ===
{{further|Fourier transform#Laplace transform}}

Let <math>f</math> be a complex-valued Lebesgue integrable function supported on <math>[0,\infty)</math>, and let <math>F(s) = \mathcal Lf(s)</math> be its Laplace transform.  Then, within the region of convergence, we have
:<math>F(\sigma + i\tau) = \int_0^\infty f(t)e^{-\sigma t}e^{-i\tau t}\,dt,</math>
which is the Fourier transform of the function <math>f(t)e^{-\sigma t}</math>.<ref>{{cite book|author=[[Laurent Schwartz]]|title=Mathematics for the physical sciences|year=1966|publisher=Addison-Wesley}}, p 224.</ref>

Indeed, the [[Fourier transform]] is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a ''real'' variable (frequency), the Laplace transform of a function is a complex function of a ''complex'' variable. The Laplace transform is usually restricted to transformation of functions of {{math|''t''}} with {{math|''t'' ≥ 0}}. A consequence of this restriction is that the Laplace transform of a function is a [[holomorphic function]] of the variable {{math|''s''}}. Unlike the Fourier transform, the Laplace transform of a [[distribution (mathematics)|distribution]] is generally a [[well-behaved]] function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a [[power series]] representation. This power series expresses a function as a linear superposition of [[moment (mathematics)|moments]] of the function. This perspective has applications in probability theory.

Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument {{math|1=''s'' = ''iω''}}<ref>{{citation
 | last = Titchmarsh | first = E. | author-link = Edward Charles Titchmarsh
 | title = Introduction to the theory of Fourier integrals
 | isbn = 978-0-8284-0324-5
 | orig-year = 1948
 | year = 1986
 | edition = 2nd
 | publisher = [[Clarendon Press]]
 | page = 6
}}</ref><ref>{{harvnb|Takacs|1953|p=93}}</ref> when the condition explained below is fulfilled,

<math display="block">\begin{align}
  \hat{f}(\omega) &= \mathcal{F}\{f(t)\} \\[4pt]
                  &= \mathcal{L}\{f(t)\}|_{s = i \omega}  =  F(s)|_{s = i \omega} \\[4pt]
                  &= \int_{-\infty}^\infty e^{-i \omega t} f(t)\,dt~.
\end{align}</math>

This convention of the Fourier transform (<math>\hat f_3(\omega)</math> in {{Section link|Fourier transform|Other_conventions}}) requires a factor of {{math|{{sfrac|1|2''π''}}}} on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the [[frequency spectrum]] of a [[signal (information theory)|signal]] or dynamical system.

The above relation is valid as stated [[if and only if]] the region of convergence (ROC) of {{math|''F''(''s'')}} contains the imaginary axis, {{math|1=''σ'' = 0}}.

For example, the function {{math|1=''f''(''t'') = cos(''ω''<sub>0</sub>''t'')}} has a Laplace transform {{math|1=''F''(''s'') =  ''s''/(''s''<sup>2</sup> + ''ω''<sub>0</sub><sup>2</sup>)}} whose ROC is {{math|Re(''s'') > 0}}. As {{math|1=''s'' = ''iω''<sub>0</sub>}} is a pole of  {{math|''F''(''s'')}}, substituting {{math|1=''s'' = ''iω''}} in {{math|''F''(''s'')}} does not yield the Fourier transform of {{math|''f''(''t'')''u''(''t'')}}, which contains terms proportional to the [[Dirac delta functions]] {{math|''δ''(''ω'' ± ''ω''<sub>0</sub>)}}.

However, a relation of the form
<math display="block">\lim_{\sigma\to 0^+} F(\sigma+i\omega) = \hat{f}(\omega)</math>
holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a [[weak limit]] of measures (see [[vague topology]]). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of [[Paley–Wiener theorem]]s.