Cool video! I've known this result for a while, but I've never seen a proof of it. I think that it is strange that there was care to mention why we can move the derivative inside the integral using the Leibniz integral rule, but there was no mention of why we could move the limit inside the integral to compute "f(∞)". We can do this because of the Lebesgue dominated convergence theorem. The integrand e^(-t^2(1+x^2))/(1+x^2) is bounded above by e^-x and \int_0^∞ e^-x dx = 1 < ∞, so we can pass the limit inside the integral.
Choosing the right function can take some trial an error. We want a function that can be integrated after we take the derivative. The final function seems like magic but it often comes from trial and error. After enough of these, we get the sense of what might work and what might not work.
Cool video! I've known this result for a while, but I've never seen a proof of it. I think that it is strange that there was care to mention why we can move the derivative inside the integral using the Leibniz integral rule, but there was no mention of why we could move the limit inside the integral to compute "f(∞)". We can do this because of the Lebesgue dominated convergence theorem. The integrand e^(-t^2(1+x^2))/(1+x^2) is bounded above by e^-x and \int_0^∞ e^-x dx = 1 < ∞, so we can pass the limit inside the integral.
That's nifty, but how do I just pull a function like that out of the ether?
Choosing the right function can take some trial an error. We want a function that can be integrated after we take the derivative. The final function seems like magic but it often comes from trial and error. After enough of these, we get the sense of what might work and what might not work.
A typo at the end. It must be sqrt(pi)/2, not sqrt(pi/2)
Thank you!