The Chemistry Maths Book, Second Edition

436 Chapter 15Orthogonal expansions. Fourier analysis

The Fourier-series representation of the function in the finite interval is therefore

(ii) The Fourier integral

To find the value of this expression in the limitl 1 → 1 ∞, we make the substitution

y

n

1 = 1 nπ 2 l, ∆y

n

1 = 1 π 2 l

Then

(15.73)

and, in the limitl 1 → 1 ∞,

(15.74)

(the constant term, corresponding ton 1 = 10 , does not contribute).

This Fourier cosine integral representation of the function can be written in

the form (15.69) as

(15.75)

withu(y)given by (15.70). Thus

(15.76)

as required by (15.74). For the corresponding pair of Fourier transforms see

Figure 15.14(a).

The exponential form

The Fourier-integral representation of a function, equations (15.69) to (15.71), can be

written in exponential form by making use of the Euler relations

(15.77)

cosxy (e e ) sinxy ( )

i

ee

ixy ixy ixy ixy

=+, = −

−−

1

2

1

2

==

22

0

A

xy dx

Ak ky

ky

k

ππ

Z cos

sin( )

()

uy()==fx xydx()cos fx xydx()cos

−

+

12

0

ππ

ZZ

∞

∞∞

fx()=Z uy xydy()cos

0

∞

fx

Ak ky

ky

() xy dy

sin( )

()

= cos

2

0

π

Z

∞

fx

Ak

ky

ky

xy

n

n

n

n

()

sin( )

()

=+ cos

=















∑

21

2

1

π

∞











∆y

n

fx

Ak

l

nkl

nkl

nx

l

n

()

sin( )

()

=+ cos









=

∑

21

2

1

∞

π

π

π







