Physical Chemistry Third Edition

1254 B Some Useful Mathematics

We have incorporated thea 0 term into the first sum, using the fact that cos(0)1. We

have also used the fact that the integral of a sum is equal to the sum of the integrals of

the terms if the series is uniformly convergent.

We now apply the orthogonality facts to find that all of the integrals on the right-hand

side of Eq. (B-108) vanish except for the term with two cosines in whichnm. The

result is, form0:

∫L

−L

f(x)cos(mπx/L)dxamL (B-109)

or

am

1

L

∫L

−L

f(x)cos(mπx/L)dx (m0) (B-110)

To finda 0 , we use the fact that

∫L

−L

cos(0)cos(0)dx

∫L

−L

dx 2 L (B-111)

which leads to

a 0 

1

2 L

∫L

−L

f(x)dx (B-112)

A similar procedure consisting of multiplication by sin(mπx/L) and integration from

−LtoLyields

bm

1

L

∫L

−L

f(x)sin(mπx/L)dx (B-113)

A function must be integrable in order to be represented by a Fourier series, but it

does not have to be continuous. It can have step discontinuities, as long as the step in

the function is finite. At a step discontinuity, a Fourier series will converge to a value

halfway between the value just to the right of the discontinuity and the value just to

the left of the discontinuity.

If the functionf(x) is an even function ofx, all of thebncoefficients will vanish,

and only the cosine terms will appear in the series. Such a series is called aFourier

cosine series.Iff(x) is an odd function ofx, only the sine terms will appear, and the

series is called aFourier sine series. We can represent a function that is not necessarily

periodic by a Fourier series if we are only interested in representing the function in

the interval−L<x<L. The Fourier series will be periodic with period 2L, and the

series will be equal to the function inside the interval, but not necessarily equal to the

function outside the interval. If we want to represent a function only in the interval

0 <x<Lwe can regard it as the right half of an odd function or the right half of an

even function, and can therefore represent it either with a sine series or a cosine series.

These two series would have the same value in the interval 0<x<Lbut would be

the negatives of each other in the interval−L<x<0.

It is a necessary condition for the convergence of Fourier series that the coeffi-

cients become smaller and smaller and approach zero asnbecomes larger and larger.

If a Fourier series is convergent, it will be uniformly convergent for all values ofx.