Fourier transforms are used in many branches of physics and engineering
to relate functions described in terms of conjugate variables. These variables
are those such as wavevector and position, or frequency and time, which
are associated with waves and particles respectively and thereby obey a
version of the uncertainty principle. In quantum physics these relations
are extended to position and momentum, or energy and time, via the introduction
of Planck's constant h in p = (h/2p)
k and E = hn.
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 twoslit diffraction pattern. 
Some of the most obvious 1D transforms are

Latest version of this document: 25th January 2005.