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A result linking the asymptotic behaviour of a function near  $  0 $
with the asymptotic behaviour of its [[Laplace transform|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====
<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>