Physical Chemistry Third Edition

1256 B Some Useful Mathematics

integrals converge. For the integral of Eq. (B-118) to converge, the following integral

must converge:

∫∞

−∞

|f(x)|^2 dx <∞ (B-119)

We say that the functionf(x) must besquare integrable. The functionf(x) must

approach zero asx→−∞and asx→∞to be square integrable. If the Fourier

transformF(k) exists, it will also be square integrable.

If the functionf(x) is an even function, its Fourier transform is aFourier cosine

transform:

F(k)

1

√

2 π

∫∞

−∞

f(x)cos(kx)dx

√

2

π

∫∞

0

f(x)cos(kx)dx (B-120)

The second version of the transform is called aone-sided cosine transform.Iff(x)is

an odd function, its Fourier transform is aFourier sine transform:

F(k)

1

√

2 π

∫∞

−∞

f(x)sin(kx)dx

√

2

π

∫∞

0

f(x)sin(kx)dx (B-121)

There is a useful theorem for the Fourier transform of a product of two func-

tions, called theconvolution theoremor theFaltung theorem(Faltungis German for

“folding”). Theconvolutionof two functionsf(x) andg(x) is defined as the integral

1

√

2 π

∫∞

−∞

f(x)g(x−y)dy (B-122)

This integral is a function ofx, and its Fourier transform is equal toF(k)G(k) where

F(k) is the Fourier transform off(x) andG(k) is the Fourier transform ofg(x).^1

Since the Fourier transform is nearly the same going in both directions, the analogous

convolution

1

√

2 π

∫∞

−∞

F(1)G(k−1)dl (B-123)

has as its Fourier transform the productf(x)g(x).

(^1) Philip M. Morse and Herman Feshbach,Methods of Theoretical Physics, Part I, McGraw-Hill,
New York, 1953, p. 464ff.