In this question, we study the case $\lambda(t) = t^2$ and $f(t) = \dfrac{1}{1+t^2}$ for all $t \in \mathbb{R}^+$. II.C.1) Determine $E$. What is the value of $Lf(0)$? II.C.2) Prove that $Lf$ is differentiable. II.C.3) Show the existence of a constant $A > 0$ such that for all $x > 0$, we have $$Lf(x) - (Lf)^{\prime}(x) = \frac{A}{\sqrt{x}}.$$ II.C.4) We denote $g(x) = e^{-x}Lf(x)$ for $x \geqslant 0$. Show that for all $x \geqslant 0$, we have $$g(x) = \frac{\pi}{2} - A\int_0^x \frac{e^{-t}}{\sqrt{t}}\,dt.$$ II.C.5) Deduce from this the value of the integral $\displaystyle\int_0^{+\infty} e^{-t^2}\,dt$.
In this question, we study the case $\lambda(t) = t^2$ and $f(t) = \dfrac{1}{1+t^2}$ for all $t \in \mathbb{R}^+$.
II.C.1) Determine $E$. What is the value of $Lf(0)$?
II.C.2) Prove that $Lf$ is differentiable.
II.C.3) Show the existence of a constant $A > 0$ such that for all $x > 0$, we have
$$Lf(x) - (Lf)^{\prime}(x) = \frac{A}{\sqrt{x}}.$$
II.C.4) We denote $g(x) = e^{-x}Lf(x)$ for $x \geqslant 0$.
Show that for all $x \geqslant 0$, we have
$$g(x) = \frac{\pi}{2} - A\int_0^x \frac{e^{-t}}{\sqrt{t}}\,dt.$$
II.C.5) Deduce from this the value of the integral $\displaystyle\int_0^{+\infty} e^{-t^2}\,dt$.