Let $f \in \mathcal { C } ^ { 0 } ( [ a , b ] )$ such that $f ( a ) \neq 0$ and $\varphi \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. For every parameter $t \in \mathbb { R }$, we denote $$F ( t ) = \int _ { a } ^ { b } e ^ { - t \varphi ( x ) } f ( x ) d x$$ Case where the phase $\varphi$ has no critical point in $[ a , b ]$. We assume that $\varphi ^ { \prime } ( x ) > 0$ for all $x \in [ a , b ]$. (a) Show that $\Phi : x \mapsto \varphi ( x ) - \varphi ( a )$ is a bijection from $[ a , b ]$ onto an interval of the form $[ 0 , \beta ]$, and that it is of class $\mathcal { C } ^ { 1 }$. (b) Show that $$F ( t ) \underset { t \rightarrow + \infty } { \sim } \frac { e ^ { - t \varphi ( a ) } f ( a ) } { \varphi ^ { \prime } ( a ) t }$$ Hint. One can reduce to the case treated in question 1a) using a change of variable.
Let $f \in \mathcal { C } ^ { 0 } ( [ a , b ] )$ such that $f ( a ) \neq 0$ and $\varphi \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. For every parameter $t \in \mathbb { R }$, we denote
$$F ( t ) = \int _ { a } ^ { b } e ^ { - t \varphi ( x ) } f ( x ) d x$$
Case where the phase $\varphi$ has no critical point in $[ a , b ]$. We assume that $\varphi ^ { \prime } ( x ) > 0$ for all $x \in [ a , b ]$.\\
(a) Show that $\Phi : x \mapsto \varphi ( x ) - \varphi ( a )$ is a bijection from $[ a , b ]$ onto an interval of the form $[ 0 , \beta ]$, and that it is of class $\mathcal { C } ^ { 1 }$.\\
(b) Show that
$$F ( t ) \underset { t \rightarrow + \infty } { \sim } \frac { e ^ { - t \varphi ( a ) } f ( a ) } { \varphi ^ { \prime } ( a ) t }$$
Hint. One can reduce to the case treated in question 1a) using a change of variable.