Fourier transform of exp( -pi*t^2) ????

Question

please show me the way how it is calculated .i guess the naswer is exp(-pi*f^2) .

Answer 1

Not an easy thing to show, as the source below indicates.

Answer 2

First, there are several constants that can vary from book to book when talking about the Fourier transform. Different choices of those constants will give slightly different answers.

There are a couple of ways of seeing this. Perhaps the easiest is to notice that if y=exp(-pi t^2), then y'+pi*t*y=0. Let F(x) be the fourier transform of y(t). Then the transform of y' will be pi*x*F(x) and the transform of pi*t*y will be F'(x), so the transform of the differential equation satisfied by y is exactly the same as the original. Thus F is a constant multiple of y. By looking at F(0), which is the improper integral of y, you can determine that the constant is 1.

The other way is to write out the Fourier integral and complete the square in the exponential. This allows a quadratic exponential to be factored out. The remaining integral can be shown to not depend on x by looking at complex integrals around curves.