Fourier SeriesArunn Narasimhan
Some notes from “The Analytical Theory of Heat” by Jean Baptiste Joseph Fourier, an abridged version of which I am reading now.
The series which goes by his name nowadays appeared in Chapter 3 of Fourier’s classic “The Analytical Theory of Heat”, which appeared in print in 1822 for the first time.
If there is a function f(x) then Fourier tells us that it can be defined within the range [-∞, +∞] using an infinite summation of the trigonometric functions sin(x) and cos(x).
In an equation form it would be represented as
We know f(x) represents a curve in the x-f(x) space. Suppose, if f(x) = mx, we know the curve is a straight line. And we know that this mx can be easily represented using the RHS of Eq. (1) if we can suitably “tweak” the constants. Similarly, if f(x) is sin(x) or cos(x) or ex or simply 1, we know how to tweak the constants on the RHS of Eq. (1) to make it work. But what Fourier did was, for an arbitrary function f(x), he proved that the series representation in Eq. (1) holds true and also showed a way to find the constants as
That is, to find the coefficient a0, Fourier integrates Eq. (1) on both sides within the range [-∞, +∞] to obtain Eq. (2); to obtain an, he multiplies Eq. (1) first on both sides with cos(mx) and then integrates over the range [-π, +π] to obtain Eq. (3) and follows the same procedure for bn by using the sin(mx) as the multiplication factor to obtain Eq. (4).
Two assumptions that Fourier made in the above scheme are the integrability of f(x) (else one wouldn’t know the answers for the coefficients) and the integral of the infinite sums was identical to the infinite sum of the integrals as follows
Daniel Bernoulli a century earlier in 1740s has proposed such an infinite series expression as in Eq. (1) while developing a theory for vibrating musical strings but was not able to find the coefficients (Eqs. 2 to 4). According to Stephen Hawking , the determination of the coefficients (Eqs. 2 to 4) was the greatest accomplishment of Fourier. According to Lord Kelvin,
Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it is said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics…Fourier is a mathematical poem.
[The above quote was taken from the recent compilation by Stephen Hawking of the works of thirty greatest mathematicians according to him]
Just to give an example, if f(x) is such that it represents the periodic triangle like curve depicted in white color in the figure below and the red color curve, the result of using Eq. (1) to approximate the actual function f(x).
Using a pair of sine and cosine (in the infinite summation of RHS of Eq. (1)) would get us to this figure already
And with about 13 to 14 sets of sine and cosines, we can obtain with fair excatitude, the required triangle like curve as follows.
In other words, a few suitable summation of wiggly curves lead us to a rigid choppy curve - all courtesy Fourier, who showed it can be done by 1822.
You can play with this Java applet here.
Now, how did Fourier use Eq. (1) to explain heat diffusion? In a separate note.
 God Created the Integers, edited with commentary by Stephen Hawking, Penguin, 2005.