Watson lemma

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