Physical Chemistry Third Edition

B Some Useful Mathematics 1253

Property 7. The determinant of a matrix is equal to the determinant of the transpose of that

matrix.

det(A ̃)det(A) (B-102)

These properties can be verified using the expansion of a determinant by minors.

B.8 Fourier Series

Fourier series are important examples of representations of functions as linear com-

binations of basis functions. The basis functions in Fourier series are sine and cosine

functions, which are periodic functions. Aperiodic functionofxwith period 2Lhas

the property that

f(x+ 2 L)f(x) (definition of a periodic function) (B-103)

for all values ofx. A Fourier series that represents a periodic function of period 2Lis

f(x)a 0 +

∑∞

n 1

ancos

(nπx

L

)

+

∑∞

n 1

bnsin

(nπx

L

)

(B-104)

Different functions are represented by having differentaandbcoefficients.

Fourier proved the following facts about Fourier series: (1) Any Fourier series in

xis uniformly convergent for all real values ofx; (2) the set of sine and cosine basis

functions in Eq. (B-104) is acomplete setfor the representation of periodic functions

of period 2L. This means that any periodic function obeying certain conditions such as

integrability can be accurately represented by the appropriate Fourier series. It is not

necessary that the function be continuous.

To find the coefficients in a Fourier series, we use theorthogonalityof the basis

functions. Orthogonality means that ifmandnare integers,

∫L

−L

cos(mπx/L)cos(nπx/L)dxLδmn

{

L if mn 0

0ifm 0

(B-105)

∫L

−L

sin(mπx/L)sin(nπx/L)dxLδmn

{

L if mn

0ifm 0

(B-106)

∫L

−L

cos(mπx/L)sin(nπx/L)dx 0 (B-107)

To findamform0 we multiply both sides of Eq. (B-104) by cos(mπx/L) and

integrate from−LtoL.

∫L

−L

f(x)cos(mπx/L)dx

∑∞

n 0

an

∫L

−L

cos(nπx/L)cos(mπx/L)dx

+

∑∞

n 0

bn

∫L

−L

sin(nπx/L)cos(mπx/L)dx (B-108)