The first search of the web that we did under (Fourier + transform) gave 8358 entries, more than we really need for this course.... A list of the more useful sites we have found to date is given below: you could also go to the library or get out your old math books, but this is more fun.
Everyone is clear that the definition of the one dimensional (1D)
transform g(k) into f(x) is defined as
and that the inverse transform is
but the problem arises if you are picky about the position of the 2
p.
The symmetric form has 1/root(2p
)
in both equations. The other possibility is to have the 2p
in the exponent, as is used for
example in Cowley's Diffraction Physics,
chapter 2. Then one doesn't need anything up front.
Much of the time we are only interested
in shapes of the functions and in this case the 2
p's,
or in general the normalization constants, don't matter.
A related technique used to describe diffraction paterns is the
Convolution
theorem. This is described by Nimish Hathi, using the example of the
familiar two-slit diffraction pattern.