Past exam of the integration course of the University of Oxford / 2023 / 2 / b

Let $J={\int }_0^{\infty }{e}^{-x^2}dx$. It is positive and finite: on $(0,1)$ the integrand is at most one, and on $[1,\infty )$ it is at most $e^{-x}$. Since the planar integrand is nonnegative and measurable, Tonelli's theorem givesThe change of variables formula in polar coordinates, with $x=r\cos \theta$, $y=r\sin \theta$ and absolute Jacobian determinant $r$, yieldsOne can apply the formula first on $\delta <r<R$, $0<\theta <\pi /2$, where the coordinate map is a diffeomorphism, and then exhaust the quadrant using the monotone convergence theorem. Taking the positive square root proves the half-line Gaussian integral:Here $dx$ is understood in the one-dimensional integral; it is missing from the printed question.

Put $h(t,x)=e^{-x^2-t^2/x^2}$. For every $t\ge 0$, $0<h(t,x)\le e^{-x^2}$, and the upper bound is a Lebesgue integrable function by part (b)(i). Since $h(t,x)\to e^{-x^2}$ for each $x>0$ as $t\downarrow 0$, the dominated convergence theorem gives right continuity at zero andTo apply dominated differentiation under the integral sign, fix $t_0>0$ and restrict $t$ to $[t_0/2,3t_0/2]$. ThenFor $x\ge 1$, $D(x)\le 3t_0{e}^{-x^2}$. For $0<x<1$, writing $a=t_0^2/4>0$, the substitution rule $u=1/x$ givesThus $D$ is integrable. The mean value theorem bounds nearby difference quotients by $D$, so the dominated convergence theorem justifies the derivative at $t_0$. As $t_0$ was arbitrary,Now make the change of variables $u=t/x$. The reversal of limits cancels the negative differential and givesConsequently $F^{\prime }(t)=-2F(t)$. This also follows in sign from the fact that the positive integrand decreases strictly as $t$ increases: the printed equation with a positive sign cannot hold. Solving the linear differential equation, or differentiating $e^{2t}F(t)$, and using right continuity at zero givesThis is a half-line instance of the Gaussian integral with an inverse-square term.