The Chemistry Maths Book, Second Edition

430 Chapter 15Orthogonal expansions. Fourier analysis

This example shows how an arbitrary wave is

built up by interference of harmonic waves, with

constructive (additive) interference in some regions

and destructive (subtractive) interference in others.

The constant term S

1

is the mean valueof the func-

tion in the interval. The two-term approximation,

S

2

, includes the first correction to this mean value,

and S

3

, S

4

, =include successively smaller correc-

tions. The graph of the partial sum S

50

is shown in Figure 15.8, and demonstrates

clearly how the Fourier series, continuous at all points in the interval, deals with the

discontinuities of the function.

0 Exercises 8–11

Change of period

A periodic functionf(x)with period 2 πis transformed into a periodic functiong(z)

with period 2 lwhen the variable xinf(x)is replaced byπz 2 l. Thus, if

then

For example, the functionsin 1 x 1 = 1 sin(πz 2 l)is periodic in xwith period 2π, and is

periodic in zwith period 2 l.

This simple change of variable is used to generalize the Fourier method, equations

(15.37) to (15.39), for the expansion of functions with periods other than 2π. Thus, a

functionf(x)that is periodic inxwith period 2 l,

f(x) 1 = 1 f(x 1 + 12 l) (15.47)

can be expanded as the Fourier series

(15.48)

with Fourier coefficients

(15.49)

(15.50) b

l

fx

nx

l

dx

n

l

=

−

+

1

Z ()sin

π

a

l

fx

nx

l

dx

n

l

=

−

+

1

Z ()cos

π

fx

a

nx

l

b

nx

l

n

nn

()=+ cos +sin













=

∑

0

1

2

∞

ππ

fx f

l

( ) ()()+= zl gzl+











222 π =+

π

fx f

z

l

()= gz()











=

π

1

−π +π

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Figure 15.8