We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$. If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Deduce the existence and value of $\lim _ { N \rightarrow + \infty } \int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t$ in the case where $a < b$.
We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$.\\
If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$\\
Deduce the existence and value of $\lim _ { N \rightarrow + \infty } \int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t$ in the case where $a < b$.