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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100701.png" /> be a [[Function of bounded variation|function of bounded variation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100702.png" />, for all positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100703.png" />. The [[Integral|integral]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100704.png" /></td> </tr></table>
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Let  $  G ( t ) $
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be a [[Function of bounded variation|function of bounded variation]] on  $  0 \leq  t \leq  R $,
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for all positive  $  R $.
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The [[Integral|integral]]
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$$
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f ( s ) = \int\limits _ { 0 } ^  \infty  {e ^ {- st } }  {dG ( t ) } = {\lim\limits } _ {R \rightarrow \infty } \int\limits _ { 0 } ^ { R }  {e ^ {- st } }  {dG ( t ) }
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$$
  
 
is known as a (formal) Laplace–Stieltjes integral.
 
is known as a (formal) Laplace–Stieltjes integral.
  
If the integral converges for some complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100705.png" />, then it converges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100706.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100707.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100708.png" /> is then the Laplace–Stieltjes transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l1100709.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007010.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007011.png" /> for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007013.png" /> that is Lebesgue integrable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007014.png" /> (see [[Lebesgue integral|Lebesgue integral]]), then the Laplace–Stieltjes transform becomes the [[Laplace transform|Laplace transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007016.png" />.
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If the integral converges for some complex number $  s _ {0} $,  
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then it converges for all $  s $
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with  $  { \mathop{\rm Re} } ( s ) > { \mathop{\rm Re} } ( s _ {0} ) $,  
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and the function $  f ( s ) $
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is then the Laplace–Stieltjes transform of $  G $.  
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If $  G $
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is of the form $  G ( t ) = \int _ {0}  ^ {t} {g ( t ) }  {dt } $
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for a function $  g $
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on $  [ 0,t ] $
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that is Lebesgue integrable for all $  t $(
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see [[Lebesgue integral|Lebesgue integral]]), then the Laplace–Stieltjes transform becomes the [[Laplace transform|Laplace transform]] $  f ( s ) = \int _ {0}  ^  \infty  {e ^ {- st } g ( t ) }  {dt } $
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of $  g $.
  
There is also a corresponding two-sided Laplace–Stieltjes transform (or bilateral Laplace–Stieltjes transform) for suitable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110070/l11007017.png" />.
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There is also a corresponding two-sided Laplace–Stieltjes transform (or bilateral Laplace–Stieltjes transform) for suitable functions $  G $.
  
 
See [[Laplace transform|Laplace transform]] for additional references.
 
See [[Laplace transform|Laplace transform]] for additional references.

Latest revision as of 22:15, 5 June 2020


Let $ G ( t ) $ be a function of bounded variation on $ 0 \leq t \leq R $, for all positive $ R $. The integral

$$ f ( s ) = \int\limits _ { 0 } ^ \infty {e ^ {- st } } {dG ( t ) } = {\lim\limits } _ {R \rightarrow \infty } \int\limits _ { 0 } ^ { R } {e ^ {- st } } {dG ( t ) } $$

is known as a (formal) Laplace–Stieltjes integral.

If the integral converges for some complex number $ s _ {0} $, then it converges for all $ s $ with $ { \mathop{\rm Re} } ( s ) > { \mathop{\rm Re} } ( s _ {0} ) $, and the function $ f ( s ) $ is then the Laplace–Stieltjes transform of $ G $. If $ G $ is of the form $ G ( t ) = \int _ {0} ^ {t} {g ( t ) } {dt } $ for a function $ g $ on $ [ 0,t ] $ that is Lebesgue integrable for all $ t $( see Lebesgue integral), then the Laplace–Stieltjes transform becomes the Laplace transform $ f ( s ) = \int _ {0} ^ \infty {e ^ {- st } g ( t ) } {dt } $ of $ g $.

There is also a corresponding two-sided Laplace–Stieltjes transform (or bilateral Laplace–Stieltjes transform) for suitable functions $ G $.

See Laplace transform for additional references.

References

[a1] D.V. Widder, "An introduction to transform theory" , Acad. Press (1971)
How to Cite This Entry:
Laplace-Stieltjes transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace-Stieltjes_transform&oldid=15100
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article