Editing Laplace transform (section)

== Relationship to other transforms ==

=== Laplace–Stieltjes transform ===
The (unilateral) Laplace–Stieltjes transform of a function {{math|''g'' : ℝ → ℝ}} is defined by the [[Lebesgue–Stieltjes integral]]

<math display=block> \{ \mathcal{L}^*g \}(s) = \int_0^\infty e^{-st} \, d\,g(t) ~.</math>

The function {{math|''g''}} is assumed to be of [[bounded variation]].  If {{math|''g''}} is the [[antiderivative]] of {{math|''f''}}:

<math display=block> g(x) = \int_0^x f(t)\,d\,t </math>

then the Laplace–Stieltjes transform of {{mvar|g}} and the Laplace transform of {{mvar|f}} coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the [[Stieltjes measure]] associated to {{mvar|g}}. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its [[cumulative distribution function]].<ref>{{harvnb|Feller|1971|p=432}}</ref>

=== 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.

=== Mellin transform ===
{{Main|Mellin transform}}
The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables.

If in the Mellin transform
<math display=block>G(s) = \mathcal{M}\{g(\theta)\} = \int_0^\infty \theta^s g(\theta) \, \frac{d\theta} \theta </math>
we set {{math|1=''θ'' = ''e''<sup>−''t''</sup>}} we get a two-sided Laplace transform.

=== Z-transform ===
{{further|Z-transform#Relationship to Laplace transform}}

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of
<math display=block> z \stackrel{\mathrm{def} }{ {}={} } e^{sT} ,</math>
where {{math|1=''T'' = 1/''f<sub>s</sub>''}} is the [[sampling interval]] (in units of time e.g., seconds) and  {{math|''f<sub>s</sub>''}} is the [[sampling rate]] (in [[samples per second]] or [[hertz]]).

Let
<math display=block> \Delta_T(t) \ \stackrel{\mathrm{def}}{=}\  \sum_{n=0}^{\infty}  \delta(t - n T) </math>
be a sampling impulse train (also called a [[Dirac comb]]) and
<math display=block>\begin{align}
  x_q(t)   &\stackrel{\mathrm{def} }{ {}={} }  x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty}  \delta(t - n T) \\
           &= \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T)
\end{align}</math>
be the sampled representation of the continuous-time {{math|''x''(''t'')}}
<math display=block> x[n] \stackrel{\mathrm{def} }{ {}={} }  x(nT) ~.</math>

The Laplace transform of the sampled signal {{math|''x''<sub>''q''</sub>(''t'') }} is
<math display=block>\begin{align}
  X_q(s) &= \int_{0^-}^\infty x_q(t) e^{-s t} \,dt \\
         &= \int_{0^-}^\infty \sum_{n=0}^\infty x[n] \delta(t - n T) e^{-s t} \, dt \\
         &= \sum_{n=0}^\infty x[n] \int_{0^-}^\infty \delta(t - n T) e^{-s t} \, dt \\
         &= \sum_{n=0}^\infty x[n] e^{-n s T}~.
\end{align}</math>

This is the precise definition of the unilateral Z-transform of the discrete function {{math|''x''[''n'']}}

<math display=block> X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} </math>
with the substitution of {{math|''z'' → ''e''<sup>''sT''</sup>}}.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,
<math display=block>X_q(s) = X(z) \Big|_{z=e^{sT}}.</math>

The similarity between the Z- and Laplace transforms is expanded upon in the theory of [[time scale calculus]].

=== Borel transform ===
The integral form of the [[Borel summation|Borel transform]]
<math display=block>F(s) = \int_0^\infty f(z)e^{-sz}\, dz</math>
is a special case of the Laplace transform for {{math|''f''}} an [[entire function]] of exponential type, meaning that
<math display=block>|f(z)|\le Ae^{B|z|}</math>
for some constants {{math|''A''}} and {{math|''B''}}.  The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. [[Nachbin's theorem]] gives necessary and sufficient conditions for the Borel transform to be well defined.

=== Fundamental relationships ===
Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.