1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

114 Chapter 1 Fourier Series and Integrals

This is the complex form of the Fourier series forf.Itiseasytoderivethe
universal formula

cn=^1

2 π

∫π

−π

f(x)e−inxdx, (2)

which is valid for all integersn, positive, negative, or zero. The complex form is
used especially in physics and electrical engineering. Sometimes the function
corresponding to a Fourier series can be recognized by use of the complex
form.

Example.
The series

∑∞

n= 1

(− 1 )n+^1

n cos(nx)

may be considered the real part of

∑∞

n= 1

(− 1 )n+^1

n e

inx=

∑∞

n= 1

(− 1 )n+^1

n

(

eix

)n

(3)

because the real part ofeiθis cos(θ ). The series on the right in Eq. (3) is recog-
nized as a Taylor series,

∑∞

n= 1

(− 1 )n+^1

n

(

eix

)n

=ln

(

1 +eix

)

.

Some manipulations yield

1 +eix=eix/^2

(

eix/^2 +e−ix/^2

)

= 2 eix/^2 cos

(

x

2

)

,

ln

(

1 +eix

)

=ix 2 +ln

(

2cos

(

x

2

))

.

The real part of ln( 1 +eix)is ln(2cos(x/ 2 ))when−π<x<π.Thus,wederive
the relation

ln

(

2cos

(x

2

))

∼

∑∞

n= 1

(− 1 )n+^1

n

cos(nx), −π<x<π. (4)

(The series actually converges except atx=±π,± 3 π,....)