In the field of quantum mechanics the Fourier transform shows the relationship between momentum/wave-number space, phi(k), and position space psi(x). Equations 1 and 2 below are a typical example of this relationship:
Let's see if we can derive equations 1 and 2 from scratch. We start with perhaps the simplest transforms:
At equation 3 above, we make the right side a function of x by multiplying by e^ikx. If we multiply both sides by e^-ikx, we get equation 4 and equation 4's right side transforms from a function of x to a function of k.
Next, we take equation 3 and find the integral of both sides with respect to k:
We solve the integral on the left side first.
It looks like we are going to get infinity. Darn! We want something finite. Here's what we are dealing with: Imagine a line segment with point zero at the center. The furthest point to the right is an infinite number of points from point zero. Going to the left, there are a minus infinite number of points.
Suppose we bend the line segment into a half circle like this:
We deduce that pi/2 radians is equivalent to an infinite number of points.
We take the square root of both sides of 7 to get 8:
It is most convenient that the square root of infinity is still infinity and the square root of pi/2 radians is still equivalent to an infinite number of points from zero to (pi/2)^.5. To make the math less cluttered we do the following:
Using some high-school algebra we derive equation 14 below:
By repeating the steps above we derive equation 2 (aka: equation 24):