We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $F$ the function defined by: $$F ( x ) = \int _ { 0 } ^ { + \infty } e ^ { - t / x } f ( t ) d t$$ Show that $F$ is well defined on $] 0 , + \infty [$ and that it satisfies the following asymptotic formula: $$F ( x ) = \sum _ { k = 0 } ^ { N } a _ { k } \Gamma \left( \frac { k + \lambda } { \mu } \right) x ^ { ( k + \lambda ) / \mu } + o \left( x ^ { ( N + \lambda ) / \mu } \right) \quad \text { when } x \rightarrow 0 ^ { + } .$$
We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties:
(a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$,
(b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that
$$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$
We denote $F$ the function defined by:
$$F ( x ) = \int _ { 0 } ^ { + \infty } e ^ { - t / x } f ( t ) d t$$
Show that $F$ is well defined on $] 0 , + \infty [$ and that it satisfies the following asymptotic formula:
$$F ( x ) = \sum _ { k = 0 } ^ { N } a _ { k } \Gamma \left( \frac { k + \lambda } { \mu } \right) x ^ { ( k + \lambda ) / \mu } + o \left( x ^ { ( N + \lambda ) / \mu } \right) \quad \text { when } x \rightarrow 0 ^ { + } .$$