Watson's lemma: Difference between revisions
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Bluemaster (talk | contribs) An example added |
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'''Watson's lemma''' is a result in [[mathematics]], proved by [[G. N. Watson]](1918, p. 133), that has significant |
'''Watson's lemma''' is a result in [[mathematics]], proved by [[G. N. Watson]](1918, p. 133), that has significant application within the theory on the [[asymptotic behavior]] of [[exponential]] [[integrals]]. |
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== Statement of the lemma == |
== Statement of the lemma == |
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:<math>\int_0^T e^{-x t}\,\phi(t)\ dt\ \sim\ \sum_{n=0}^\infty \frac{g^{(n)}(0)\ \Gamma(\lambda+n+1)}{n!\ x^{\lambda+n+1}},\ \ (x>0,\ x\rightarrow \infty).</math> |
:<math>\int_0^T e^{-x t}\,\phi(t)\ dt\ \sim\ \sum_{n=0}^\infty \frac{g^{(n)}(0)\ \Gamma(\lambda+n+1)}{n!\ x^{\lambda+n+1}},\ \ (x>0,\ x\rightarrow \infty).</math> |
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'''Proof:''' See, for instance, {{harvtxt|Watson|1918}} for the original |
'''Proof:''' See, for instance, {{harvtxt|Watson|1918}} for the original proof or {{harvtxt|Miller|2006}} for a more recent development. |
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== Example == |
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Find a simple asymptotic approximation for |
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:<math>f(x)=\int_0^T e^{-x t}\,t^{-\frac{1}{2}}\ dt</math> |
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for large values of <math>x</math>, that is when <math>x\rightarrow\infty</math>. |
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'''Solution:''' By direct application of Watson's lemma, with <math>\lambda=-1/2</math>, <math>g(t)=1</math>, |
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so that <math>g^{n}(t)=0</math> for <math>n\neq 0</math>, it is easy to see that: |
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:<math>f(x)=\int_0^T e^{-x t}\,t^{-\frac{1}{2}}\ dt\sim \sqrt{\pi/x}\ \ \ (\mbox{as}\ x\rightarrow\infty).</math> |
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This development used the fact that <math>\Gamma(1/2)=\sqrt{\pi}</math>, a known property of the [[Gamma function]]. |
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== References == |
== References == |
Revision as of 19:25, 1 May 2010
Watson's lemma is a result in mathematics, proved by G. N. Watson(1918, p. 133), that has significant application within the theory on the asymptotic behavior of exponential integrals.
Statement of the lemma
Assume , where has an infinite number of derivatives in the neighborhood of , with , and .
Suppose, in addition, that
Then, it is true that
is finite , and the following asymptotic equivalence holds:
Proof: See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.
Example
Find a simple asymptotic approximation for
for large values of , that is when .
Solution: By direct application of Watson's lemma, with , , so that for , it is easy to see that:
This development used the fact that , a known property of the Gamma function.
References
- Miller, P.D. (2006), Applied Asymptotic Analysis, Providence, RI: American Mathematical Society, p. 467, ISBN 9780821840788.
- Watson, G. N. (1918), "The harmonic functions associated with the parabolic cylinder", Proc. London Math. Soc., vol. 2, no. 17, pp. 116–148.