The Fourier Transform and its Applications
One of the fun ones, and this may or may not be covered by the link, is generating the &quot;cumulants&quot; for probability density functions. The basic idea in (basic, continuous) probability theory is that you have these things called &quot;random variables&quot; X,Y,Z which now need to be thought of taking on values with various probabilities; and the way we do that is to use calculus (where the d- in &quot;dx&quot; is a special notation coming from the word &quot;difference&quot;; &quot;dx&quot; means &quot;a tiny little change in x&quot;). The basic idea is that there is some <i>joint density</i> j(x, y, z) such that the probability that simultaneously x &lt; X &lt; x + dx, AND y &lt; Y &lt; y + dy, AND z &lt; Z &lt; z + dz, is simply given by j(x, y, z) dx dy dz.<p>With a bit of calculation you conclude that if we want to analyze Z = X + Y, then the probability-density for Z alone must be h(z) = ∫ dx j(x, z - x); there are a couple ways to do this but my favorite is to use Z = X + Y as a schematic to write a Dirac δ-function j(x, y, z) = j(x, y) δ(z - x - y), then the standard &quot;what is the probability-density for Z alone? integrate over both x and y!&quot; rule comes into play. But you can also reason it out by hand, if you want.<p>Anyway, then you come up with this cool definition of independence; X and Y are independent if j(x, y) factors nicely into f(x) g(y); independent probabilities tend to multiply. So that&#x27;s cool.<p>Now here&#x27;s where the Fourier transform comes in; consider using a convention where we denote the Fourier transform and with the same function-name but using square brackets rather than parentheses for its application,<p>f[q] = ℱ{p → q} f(p) = ∫ dp exp(-i p q) f(p).<p>When we Fourier-transform h(z) for Z = X + Y we get:<p>h[a] = ∫ dz exp(-i a z) h(z) = ∫ dx dy exp(-i a (x + y)) j(x, y)<p>and if the two are independent random variables we find that h[a] = f[a] g[a]; the sum of independent random variables has a product of Fourier transforms precisely because the above sum takes the form of a convolution.<p>Taking a logarithm, we have that the χ(a) = log h[a] = log f[a] + log g[a] and any properties of the logarithm of the Fourier transform of the density function must be additive amongst independent random variables. And this gives the <i>cumulants</i> of the random variables: pseudo-linear expressions (linear in independent random variables f(X + Y) = f(X) + f(Y), with f(k X) = q(k) * f(X) for some fixed q) which characterize the different &quot;moments&quot; of the random variables in question. In fact if we look carefully at the definition h[a] = ∫ dz exp(-i a z) h(z)  we see that the variable P = k Z, having distribution b(p) = h(p&#x2F;k) &#x2F; k, generates b[a] = h[k a].<p>Calculating the zeroth, χ(0) = log h(0) = log 1 = 0. No biggie. Continuing the Maclaurin expansion gives χ&#x27;(a) = h&#x27;(a)&#x2F;h(a) and at a=0 that&#x27;s -i E(Z). We knew that the expectation value was linear, so no surprise there. The next term is (-i)^2 [h&#x27;&#x27;(a) h(a) - h&#x27;(a) h&#x27;(a)]&#x2F;[h(a) h(a)] which works out to just E(Z^2) - E(Z)^2, the familiar expression for the variance; so variance is pseudolinear (in this case q(k) = k^2). The point is, then you can just go on. There are in fact infinitely many pseudolinear cumulants which scale like q(k) = k^n, coming from this Maclaurin series, with a pattern based in fact entirely on the pattern of derivatives that comes out of log(f(x)).<p>As a freebie you have a proof of the central limit theorem. Take N independent identically-distributed random variables with characteristic functions χ(a), call them each X_i, form their sum S = sum_i X_i &#x2F; N, and from the properties we&#x27;ve already seen in this comment its characteristic function must be ψ(a) = N χ(a &#x2F; N). Do the Maclaurin series to 2nd order and ignore the latter terms for large N; you get convergence to a parabola, which means that the Fourier transform of the probability density is... a Gaussian! And the inverse Fourier transform of that Gaussian is just another Gaussian, so for large N the result must be a normal random variable with mean μ and variance σ²&#x2F;N, which of course is what they must be, because means are linear and variances are pseudolinear.