Looking At A Fourier Series English Language Essay

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In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768-1830) for the purpose of solving the heat equation in a metal plate.

The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems.

The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics[1], etc.

Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. He applied this technique to find the solution of the heat equation, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides and 1811, and publishing his Théorie analytique de la chaleur in 1822.

From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.


In this section, Æ’(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2Ï€, which is to say that Æ’(x + 2Ï€) = Æ’(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2Ï€-periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−π, Ï€], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

Fourier's formula for 2Ï€-periodic functions using sines and cosines

For a periodic function Æ’(x) that is integrable on [−π, Ï€], the numbers

a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0


b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1

are called the Fourier coefficients of Æ’. One introduces the partial sums of the Fourier series for Æ’, often denoted by

(S_N f)(x) = \frac{a_0}{2} + \sum_{n=1}^N \, [a_n \cos(nx) + b_n \sin(nx)], \quad N \ge 0.

The partial sums for Æ’ are trigonometric polynomials. One expects that the functions SN Æ’ approximate the function Æ’, and that the approximation improves as N tends to infinity. The infinite sum

\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]

is called the Fourier series of Æ’.

The Fourier series does not always converge, and even when it does converge for a specific value x0 of x, the sum of the series at x0 may differ from the value Æ’(x0) of the function. It is one of the main questions in harmonic analysis to decide when Fourier series converge, and when the sum is equal to the original function. If a function is square-integrable on the interval [−π, Ï€], then the Fourier series converges to the function at almost every point. In engineering applications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counter-examples to this presumption. In particular, the Fourier series converges absolutely and uniformly to Æ’(x) whenever the derivative of Æ’(x) (which may not exist everywhere) is square integrable.[3] See Convergence of Fourier series.

It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.

Example: a simple Fourier series

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Plot of a periodic identity function-a sawtooth wave

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Animated plot of the first five successive partial Fourier series

We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave

f(x) = x, \quad \mathrm{for } -\pi < x < \pi,

f(x + 2\pi) = f(x), \quad \mathrm{for } -\infty < x < \infty.

In this case, the Fourier coefficients are given by

\begin{align} a_n &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0, \quad n \ge 0. \\ b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = -\frac{2}{n}\cos(n\pi) + \frac{2}{n^2\pi}\sin(n\pi) = 2 \, \frac{(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}

It can be proved that the Fourier series converges to Æ’(x) at every point x where Æ’ is differentiable, and therefore:

\begin{align} f(x) &= \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] \\ &=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for} \quad x - \pi \notin 2 \pi \mathbf{Z}. \end{align}






When x = Ï€, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of Æ’ at x = Ï€. This is a particular instance of the Dirichlet theorem for Fourier series.

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Heat distribution in a metal plate, using Fourier's method

One notices that the Fourier series expansion of our function looks much less simple than the formula Æ’(x) = x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures Ï€ meters, with coordinates (x, y) ∈ [0, Ï€] Ã- [0, Ï€]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by y = Ï€, is maintained at the temperature gradient T(x, Ï€) = x degrees Celsius, for x in (0, Ï€), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by

T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.

Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of  Eq.1 by sinh(ny)/sinh(nÏ€). While our example function f(x) seems to have a needlessly complicated Fourier series, the heat distribution T(x, y) is nontrivial. The function T cannot be written as a closed-form expression. This method of solving the heat problem was only made possible by Fourier's work.

Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.