Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.7 COMPLEX FOURIER SERIES

where the Fourier coefficients are given by

cr=

1

L

∫x 0 +L

x 0

f(x)exp

(

−

2 πirx

L

)

dx. (12.10)

This relation can be derived, in a similar manner to that of section 12.2, by mul-

tiplying (12.9) by exp(− 2 πipx/L) before integrating and using the orthogonality

relation

∫x 0 +L

x 0

exp

(

−

2 πipx

L

)

exp

(

2 πirx

L

)

dx=

{

L forr=p,

0forr=p.

The complex Fourier coefficients in (12.9) have the following relations to the real

Fourier coefficients:

cr=^12 (ar−ibr),

c−r=^12 (ar+ibr).

(12.11)

Note that iff(x)isrealthenc−r=c∗r, where the asterisk represents complex

conjugation.

Find a complex Fourier series forf(x)=xin the range− 2 <x< 2.

Using (12.10), forr=0,

cr=

1

4

∫ 2

− 2

xexp

(

−

πirx

2

)

dx

=

[

−

x

2 πir

exp

(

−

πirx

2

)] 2

− 2

+

∫ 2

− 2

1

2 πir

exp

(

−

πirx

2

)

dx

=−

1

πir

[exp(−πir)+exp(πir)]+

[

1

r^2 π^2

exp

(

−

πirx

2

)] 2

− 2

=

2 i

πr

cosπr−

2 i

r^2 π^2

sinπr=

2 i

πr

(−1)r. (12.12)

Forr= 0, we findc 0 = 0 and hence

x=

∑∞

r=r=0−∞

2 i(−1)r

rπ

exp

(

πirx

2

)

.

We note that the Fourier series derived forxin section 12.6 givesar=0forallrand

br=−

4(−1)r

πr

,

and so, using (12.11), we confirm thatcrandc−rhave the forms derived above. It is also
apparent that the relationshipc∗r=c−rholds, as we expect sincef(x)isreal.