Physical Chemistry Third Edition

B Some Useful Mathematics 1255

If convergence is fairly rapid, it is possible to approximate a Fourier series by one of

its partial sums.

The sine and cosine basis functions are closely related to complex exponential

functions, as shown in Eq. (B-63). One can write

bnsin

(nπx

L

)

+ancos

(nπx

L

)



1

2

(an−ibn)einπx/L+

1

2

(an+ibn)e−inπx/L

(B-114)

It is therefore possible to rewrite Eq. (B-104) as anexponential Fourier series:

f(x)

∑∞

n−∞

cneinπx/L (B-115)

We have incorporated the terms with negative exponents into the same sum with the

other terms by allowing the summation index to take on negative as well as positive

values. The function that is represented by a Fourier series does not have to be a real

function. However, if it is a real function, the coefficientsanandbnwill be real, so that

cnwill be complex.

B.9 Fourier Integrals (Fourier Transforms)

Fourier series are designed to represent periodic functions with period 2L. If we allow

Lto become larger and larger without bound, the values ofnπx/Lbecome closer and

closer together. We let

k

nπ

L

(B-116)

As the limitL→∞is taken,kbecomes a continuously variable quantity if the limit

n→∞is taken in the proper way. In this limit the exponential Fourier series of

Eq. (B-115) becomes an integral, which is called aFourier integralor aFourier

transform.

f(x)

1

√

2 π

∫∞

−∞

F(k)eikxdk (B-117)

where the coefficientcnin Eq. (B-115) is replaced by a function ofkthat is denoted by

F(k)/

√

2 π. The equation for determiningF(k) is analogous to Eqs. (B-109), (B-112),

and (B-113)

F(k)

1

√

2 π

∫∞

−∞

f(x)e−ikxdx (B-118)

We have introduced a factor of 1/

√

2 πin front of the integral in Eq. (B-117) in order

to have the same factor in front of this integral and the integral in Eq. (B-118).

The functionF(k) is called theFourier transformoff(x) and the functionf(x)is

also called the Fourier transform ofF(k). The functionf(x) is no longer required to

be periodic, because the period 2Lhas been allowed to become infinite. Since we now

have improper integrals, the functionsf(x) andF(k) must have properties such that the