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Difference between revisions of "Watson lemma"

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A result linking the asymptotic behaviour of a function near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971101.png" /> with the asymptotic behaviour of its [[Laplace transform|Laplace transform]] near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971102.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971103.png" /> have the asymptotic expansion
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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/w/w097/w097110/w0971104.png" /></td> </tr></table>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971105.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971106.png" /> be the Laplace transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971107.png" />,
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A result linking the asymptotic behaviour of a function near  $  0 $
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with the asymptotic behaviour of its [[Laplace transform|Laplace transform]] near  $  \infty $.  
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Let  $  f( t) $
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have the asymptotic expansion
  
<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/w/w097/w097110/w0971108.png" /></td> </tr></table>
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$$
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f( t)  \sim  \sum _ { n= } 1 ^  \infty  a _ {n} t ^ {\lambda _ {n} } ,
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\  t \rightarrow 0,
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$$
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w0971109.png" /> has a corresponding asymptotic expansion
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$  - 1 < \mathop{\rm Re} ( \lambda _ {1} ) <  \mathop{\rm Re} ( \lambda _ {2} ) < \dots $,
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and let  $  F $
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be the Laplace transform of  $  f $,
  
<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/w/w097/w097110/w09711010.png" /></td> </tr></table>
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$$
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F( p)  = \int\limits _ { 0 } ^  \infty  e ^ {- pt } f( t)  dt .
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$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097110/w09711011.png" />.
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Then  $  F $
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has a corresponding asymptotic expansion
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$$
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F( p)  \sim  \sum _ { n= } 1 ^  \infty 
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\frac{a _ {n} \lambda _ {n} ! }{p ^ {\lambda _ {n} + 1 } }
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,\ \
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| p | \rightarrow \infty ,
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$$
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$  - \pi / 2 <  \mathop{\rm arg} ( p) < \pi / 2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Davies,  "Integral transforms and their applications" , Springer  (1978)  pp. §1.3</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Davies,  "Integral transforms and their applications" , Springer  (1978)  pp. §1.3</TD></TR></table>

Revision as of 08:28, 6 June 2020


A result linking the asymptotic behaviour of a function near $ 0 $ with the asymptotic behaviour of its Laplace transform near $ \infty $. Let $ f( t) $ have the asymptotic expansion

$$ f( t) \sim \sum _ { n= } 1 ^ \infty a _ {n} t ^ {\lambda _ {n} } , \ t \rightarrow 0, $$

$ - 1 < \mathop{\rm Re} ( \lambda _ {1} ) < \mathop{\rm Re} ( \lambda _ {2} ) < \dots $, and let $ F $ be the Laplace transform of $ f $,

$$ F( p) = \int\limits _ { 0 } ^ \infty e ^ {- pt } f( t) dt . $$

Then $ F $ has a corresponding asymptotic expansion

$$ F( p) \sim \sum _ { n= } 1 ^ \infty \frac{a _ {n} \lambda _ {n} ! }{p ^ {\lambda _ {n} + 1 } } ,\ \ | p | \rightarrow \infty , $$

$ - \pi / 2 < \mathop{\rm arg} ( p) < \pi / 2 $.

References

[a1] B. Davies, "Integral transforms and their applications" , Springer (1978) pp. §1.3
How to Cite This Entry:
Watson lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Watson_lemma&oldid=16130