A Fourier transform is similar to a Fourier series except instead of a sum over discrete frequencies, it is an integral over continuous frequencies. A Fourier transform can represent any function, while a Fourier series can only represent periodic functions. Starting from our Fourier series,
f(x)=m=0∑∞(Amcos(Kmx)+Bmsin(Kmx))
turn the sum into an integral and replace the sinusoids with a complex exponential
f(x)=∫−∞∞A(K)eiKxdK.
Instead of discrete frequencies Km, we now have a continuous K ranging from −∞ to ∞. Instead of discrete amplitudes An and Bn, we have a continuous amplitude function A(K). We will solve for A(K) given f(x).
We multiply both sides by e−iK′x and integrate with respect to x. We then change the order of integration on the right side.
We will see shortly the significance of this δn(s) function. For now we integrate it over the real line.
∫−∞∞δn(s)ds=π1∫−∞∞ssin(ns)ds=π1Twice the Dirichlet integral∫−∞∞tsintdt,where t=ns,dt=nds=1.
Where the Dirichlet integral∫0∞tsintdt=2π (techniques for evaluating this integral are given on the linked page). Interestingly, we see that the integral of δn(s) does not depend on n.
We now return to equation (2). We see that
∫−∞∞ei(K−K′)xdx=n→∞lim2πδn(K−K′)=:2πδ(K−K′).
This limit does not exist; but as we have seen, the integral of δn over the reals equals 1 regardless of n. Even though δ does not converge, its integral does.
Let’s consider what this means for equation (1):
∫−∞∞f(x,0)e−iK′xdx=2π∫−∞∞A(K)δ(K−K′)dK.
Let’s quickly consider the behavior of δn. By L’Hôppital’s rule, lims→0δn(s)=n/π. When s is not near 0, the 1/s term dominates and δn(s)→0. As n→∞, δ(s) not near s=0 becomes insignificant compared to δ(s→0). Therefore, in the limit as n→∞, the only contribution to the integral is at s=0, or in this case at K=K′.
Since the only contribution to the integral is at K=K′ and δ integrates to 1, we conclude that ∫−∞∞A(K)δ(K−K′)dK=A(K′).
A(K′)=2π1∫−∞∞f(x,0)e−iK′xdx.
Note
δ is known as the Dirac delta and is defined by its properties
δ(x)=0 where x=0∫−∞∞δ(x)dx=1.
Strictly speaking, no function exists with these properties. We can instead consider δ to be the limit of a sequence of functions δn as n→∞. We see that the second property is preserved for all δn, and the first becomes true as n is very large.