Examples of problems that are easier in the infinite case than in the finite case.

I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can’t think of any good ones for now, but I’ll be sure to add some when I do.

Answer

One can compute the value of
0et2dt
exactly. This is known as the Gaussian integral, and it has its own Wikipedia page. The answer turns out to be 12π.

But one cannot do the same with
x0et2dt
because the antiderivative of the integrand is not an elementary function. This is why we gave a name to the error function erf(x)=2πx0et2dt, which also has its own Wikipedia page.

In that sense, the infinite case is easier than the finite case.


Addendum: The same phenomenon occurs for variants of this integral, in particular we can transform the integrand to evaluate ae(tb)2/(2c2)dt=2a|c|π as detailed here on Wikipedia.

Attribution
Source : Link , Question Author : Asinomás , Answer Author : Will R

Leave a Comment