## ABSTRACT

At the end of the previous chapter, we were given an arbitrary function f(x) deﬁned on 0 ≤ x ≤ L, and we wanted to know if there is a linear combination (possibly with inﬁnitely many terms) of the functions sin nπx

L that is identical to f(x) on 0 ≤ x ≤ L. That is, can we ﬁnd coeﬃcients cn, n = 1, 2, 3, . . ., such that

f(x) =

cn sin npix

L (3.1)

for 0 ≤ x ≤ L? Fourier’s aﬃrmative answer to this question in 1807, while not quite correct,

marks a pivotal moment in the history of mathematics. Indeed, this surprising and profound result not only disturbed Fourier’s mathematical contemporaries but, ultimately, rocked the very foundations of mathematical analysis. What we would like to do is to assume that (3.1) is true and actually ﬁnd

the values of the coeﬃcients cn for which it is, in fact, true. Let’s do this and worry about serious mathematical issues afterwards. (We say that we proceed formally.) To this end, we will ﬁx positive integer N , multiply both sides of (3.1) by sin Nπx

L and then integrate both sides from x = 0 to x = L, resulting in ∫ L

f(x) sin Npix

L dx =

cn

sin npix

L sin

Npix

L dx. (3.2)

Then, it is easy enough to calculate the integrals in (3.2). It turns out that

sin npix

L sin

Npix

L dx =

{ 0, if n = N , L 2 , if n = N ,

(3.3)

so (3.2) becomes

f(x) sin Npix

L dx = c1 · 0 = c2 · 0 + · · ·+ cN−1 · 0 + cN · L

2 + cN+1 · 0 + · · ·

or

cN = 2

∫ L

f(x) sin Npix

dx. (3.4)

n true. have we? Many readers will notice that we assumed the truth of (3.1) and then cal-

culated the values that the coeﬃcients must take on. This, of course, says nothing about the truth of (3.1). Further, the series in (3.1) is an inﬁnite series of functions. As we know, an inﬁnite series of constants may not even converge; an inﬁnite series of functions may, then, converge for some values of x and diverge for others. Finally, there is a more subtle problem with the above argument. We

actually skipped a step in going from (3.1) to (3.2)—we assumed that we could integrate the right side term-by-term, i.e., that

( ∞∑ n=1

cn sin npix

L sin

Npix

L

) dx =

cn

sin npix

L sin

Npix

L dx.