Q3
Integration by Substitution
Substitution to Evaluate Limit of an Integral Expression
View
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.