# Fourier Series

## Motivation

Periodic motion is a common occurrence. Sound waves (as in sound sythesizers), the seasons, and radio waves are just a few examples. A plot of wave amplitude versus time can be very complex as in (three periods of the oscillating wave are shown). Even the most complex periodic function can be expanded in sines and cosines using the Fourier series. Such a Fourier expansion provides an interpetation of the wave in terms of its elementary components. In the language of music this is called analysis in fundamentals and overtones. There are innumerable uses for the Fourier series in science. And the notion of expansion in a set of elementary functions is more general than Fourier series.

Expansion in a basis is a generalization of the Fourier series.   Quantum mechanics problems are commonly solved by expansion in a set of basis functions. The motive is the same as for Fourier series. An unknown wave function is to be expanded as a linear combination of well known wave functions and the problem is re-formulated to find the expansion coefficients. This is a powerful technique for solving complicated differential equations such as the quantum mechanical Schroedinger equation.

## Standard Fourier Problem

Let f(x) be an arbitrary piecewise continuous function on a finite interval (a,b). Then Fourier analysis of f(x) consists in finding the coeffients an and bn so that the following expansion or series "represents" the function as well as possible. If, for simplicity, the interval is (-p, p) the Fourier expansion is written
f(x) ~ F(x) = a0/2 +
S(an cos(nx) + bn sin(nx)) .  Equ(1)

The Fourier expansion is clearly periodic of period 2p: F(x+2p) = F(x). Even if the original function, f(x), is not periodic, the expansion proves useful in the interval (a,b). But if f(x) itself is periodic then its Fourier expansion, F(x), obtained on one period of the function represents it over all periods.

Several criteria for fitting the series to the function will lead to the same formulas for the expansion coefficients; namely,

 an = Equ(2) bn = Equ(3)

For instance, this choice minimizes the deviation between f(x) and the series expansion, .-->0. Another approach uses the rules of orthogonality of sines and cosines:

 = p dnn' Equ(4a) = 0 Equ(4b) = p dnn'(1+dn0) Equ(4c)

Then multiply both sides of equation (1) by either sin(nx) or cos(nx) and integrate over (-p, p) and use the orthogonality rules to find either Equ(2) or Equ(3).

## Exercises

Expand the square wave.

Consider the square wave, f(x) ={1 if -p<x<0, 0 if 0<x<p, and periodic of period 2p}. Show that the Fourier expansion coefficients are a0 = 1, an>1=0, bn = {0 if n is even, -2/np if n is odd}. Make a plot of f(x) and the Fourier sum through n=1, and through n=11. See the solution.

Vibrating String

See the classical eigenvalue problem arising from the vibrating string.

## Properties of Fourier Series

If f(x) is an odd function, f(-x) = -f(x), then the coefficients of cosine terms vanish; cosine terms are absent from the expansion.

If f(x) is an even function, f(-x) = f(x), then the coefficients of the sine terms vanish; sine terms are absent from the expansion.

If f(x) has a finite discontinuity at x=a, lim f(x --> a+) .ne. lim f(x --> a-), then the Fourier series expansion converges to (1/2)[f(a+)+f(a-)] at the discontinuity.

If f(x) is periodic on intervals of length 2a instead of length 2p, that is, if the primary interval is -a<x<a, then the Fourier series becomes

f(x) = a0/2 + Sn=1,2,... [an cos(npx/a) + bn sin(npx/a)]

where the coefficients an and bn are given by

## Problems

1. (a) Obtain the Fourier series for the function f(x) = px on the interval -1<x<1.  (b) Obtain the Fourier series for the function g(x) = p|x| on the interval -1<x<1. (c) These two series expansions converge to the same function in the interval 0<x<1. How is this possible?
2. Consider the two functions. (a) Demonstrate that F(¥,x) is the Fourier expansion of f(x). (b) What is the limit of f(x) as x approaches 0? (c) To what does F(M,x) converge as M increases without limit and x approaches zero?

## References

• T.W. Korner, Fourier Analysis [Cambridge University Press, 1989]
• D.W. Jordan and P. Smith, Mathematical Techniques [Oxford University Press, 1994]

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Created or up-dated 08/03/99   by R.D. Poshusta