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 a critical point at $a$. We now assume that $\varphi \in \mathcal { C } ^ { 2 } ( [ a , b ] )$, $\varphi ^ { \prime } ( a ) = 0 , \varphi ^ { \prime \prime } ( a ) > 0$, and $\varphi ^ { \prime } ( x ) > 0$ for all $\left. \left. x \in \right] a , b \right]$.
(a) Show that the formula $\psi ( x ) = \sqrt { \varphi ( x ) - \varphi ( a ) }$ defines a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. Calculate $\psi ^ { \prime } ( a )$.
(b) Show that $\psi$ is a bijection from $[ a , b ]$ onto an interval of the form $[ 0 , \beta ]$.
(c) Show that $$F ( t ) \underset { t \rightarrow + \infty } { \sim } \sqrt { \frac { \pi } { 2 \varphi ^ { \prime \prime } ( a ) } } \frac { e ^ { - t \varphi ( a ) } f ( a ) } { \sqrt { t } } .$$ Hint. One can reduce to the case treated in question 1b) using a change of variable.

Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$.
Under hypothesis (H), show that for all $\delta > 0$ such that $\delta < \min(x_0 - a, b - x_0)$, we have the asymptotic equivalence, as $t \rightarrow +\infty$, $$\int_a^b e^{tf(x)} \mathrm{d}x \sim \int_{x_0 - \delta}^{x_0 + \delta} e^{tf(x)} \mathrm{d}x.$$

Let $a < b$ be two real numbers and $f : [ a , b ] \rightarrow \mathbb { R }$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x _ { 0 } \in [ a , b ]$ where $f$ attains its maximum, we have $a < x _ { 0 } < b$, and $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$.
Under hypothesis $( \mathrm { H } )$, show that for all $\delta > 0$ such that $\delta < \min \left( x _ { 0 } - a , b - x _ { 0 } \right)$, we have the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim \int _ { x _ { 0 } - \delta } ^ { x _ { 0 } + \delta } e ^ { t f ( x ) } \mathrm { d } x$$

Show that $$\lim _ { n \rightarrow \infty } \sqrt { n } I _ { n } = \int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } } \mathrm {~d} u$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.