Fourier series started life as a method to solve problems about the ﬂow of heat through ordinary materials. It has grown so far that if you search our library’s data base for the keyword “Fourier” you will ﬁnd 425 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial diﬀerential equations, and a later chapter will show some of these tools at work. 5.1 Examples The power series or Taylor series is based on the idea that you can write a general function as an inﬁnite series of powers. The idea of Fourier series is that you can write a function as an inﬁnite series of sines and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization aside for now. Legendre polynomials are an important example of functions used for such expansions. An example: On the interval 0 < x < L the function x2 varies from 0 to L2 . It can be written as the series of cosines L2 4L2 + 2 x = 3 π 2 ∞ 1

(−1)n nπx cos 2 n L 2πx 1 3πx πx 1 − cos + cos − ··· L 4 L 9 L (1)

=

L2 3

−

4L2 π2

cos

To see if this is even plausible, examine successive partial sums of the series, taking one term, then two terms, etc. Sketch the graphs of these partial sums to see if they start to look like the function they are supposed to represent (left graph). The graphs of the series, using terms up to n = 5 does pretty well at representing the given function. 5 3 1

5 3

1

highest harmonic: 5

highest harmonic: 5

The same function can be written in terms of sines with another series: 2L2 x = π 2 ∞ 1

(−1)n+1 2 − 2 3 1 − (−1)n ) n π n

1

sin

nπx L

(2)

jnearing@miami.edu

James Nearing, University of Miami

2

5—Fourier Series

and again you can see how the series behaves by taking one to several terms of the series. (right graph) The graphs show the parabola y = x2 and partial sums of the two series with terms up to n = 1, 3, 5. The second form doesn’t seem to work as smoothly as the ﬁrst example, and there’s a reason for that. The sine functions all go to zero at x = L and x2 doesn’t, making it hard for the sum of sines to approximate the desired function. They can do it, but it takes a lot more terms in the series to get a satisfactory result. The series Eq. (1) has terms that go to zero as 1/n2 , while the terms in the series Eq. (2) go to zero only as 1/n. 5.2 Computing Fourier Series How do you determine the details of these series starting from the original function? For the Taylor series, the trick was to assume a series to be an inﬁnitely long polynomial and then to evaluate it (and its successive derivatives) at a point. You require that all of these values match those of the desired function at that one point. That method won’t work here. (Actually it can work here too, but only after a ridiculous amount of labor.) The idea of the procedure that works here is like one that you can use to determine the components of a vector in three dimensions. You write such a vector as A = A x x + A y y + Az z ˆ ˆ ˆ And then use the orthonormality of the basis vectors, x . y = 0 etc. Take the scalar product of ˆ ˆ the preceding equation with x. ˆ x . A = x . Ax x + Ay y + A z z = Ax . ˆ ˆ ˆ ˆ ˆ (3)

This lets you get all the components of A. There are orthogonality relations similar to the ones for x, y , and z , but for sines and ˆ ˆ ˆ cosines. Let n and m represent integers, then L

dx sin

0

mπx nπx sin = L L

0 n=m L/2 n = m

(4)

This is sort of like x . z = 0 and y . y = 1. ˆ ˆ ˆ ˆ More Examples For a simple example, take the function f (x) = 1, the constant on the interval 0 < x < L and assume that there is a series representation for f on this...