2011-02-06, 06:00 AM
e^-x^2 is not integrable directly, so a transformation has to first be made. Assume J = int(e^-x^2 dx), and x=y. Therefore:
J^2 = int(e^-x^2 dx * e^-y^2 dy)
J^2 = int(e^-(x^2+y^2) dx dy)
Transform:
J^2 = int(e^-r^2 rdr) * int(d(theta))
Usually, at this point, various sources integrate it from -inf to inf, and obtain sqrt(pi) as the solution of int(e^-x^2 dx) [-inf,inf].
My question is, why should this only work unless the function is integrated from -inf to inf? Why can't it work for any value of x or r?
J^2 = int(e^-x^2 dx * e^-y^2 dy)
J^2 = int(e^-(x^2+y^2) dx dy)
Transform:
J^2 = int(e^-r^2 rdr) * int(d(theta))
Usually, at this point, various sources integrate it from -inf to inf, and obtain sqrt(pi) as the solution of int(e^-x^2 dx) [-inf,inf].
My question is, why should this only work unless the function is integrated from -inf to inf? Why can't it work for any value of x or r?