Fourier series in complex form and Fourier integral

Fourier series in complex form and Fourier integral

0.1

Fourier series in complex form

The

Fourier series

expansion of a

Riemann integrable

real function

f

on the

interval

[

-

p

,

p

]

is

f

⁢

(

t

)

=

a

0

2

+

∑

n

=

1

∞

(

a

n

⁢

cos

⁡

n

⁢

π

⁢

t

p

+

b

n

⁢

sin

⁡

n

⁢

π

⁢

t

p

)

,

(1)

where the

coefficients

are

a

n

=

1

p

⁢

∫

-

p

p

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

π

⁢

t

p

⁢

d

⁢

t

,

b

n

=

1

p

⁢

∫

-

p

p

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

π

⁢

t

p

⁢

d

⁢

t

.

(2)

If one expresses the cosines and sines via Euler formulas (

http://planetmath.org/ComplexSineAndCosine

) with

exponential function

(

http://planetmath.org/ComplexExponentialFunction

), the series (1) attains the form

f

⁢

(

t

)

=

∑

n

=

-

∞

∞

c

n

⁢

e

i

⁢

n

⁢

π

⁢

t

p

.

(3)

The coefficients

c

n

could be obtained of

a

n

and

b

n

, but they are comfortably derived directly by multiplying the equation (3) by

e

-

i

⁢

m

⁢

π

⁢

t

p

and integrating it from

-

p

to

p

.  One obtains

c

n

=

1

2

⁢

p

∫

-

p

p

f

(

t

)

e

-

i

⁢

n

⁢

π

⁢

t

p

d

t

(

n

=

0

,

±

1

,

±

2

,

…

)

.

(4)

We may say that in (3),

f

⁢

(

t

)

has been dissolved to sum of

harmonics

(

elementary

waves)

c

n

⁢

e

i

⁢

n

⁢

π

⁢

t

p

with amplitudes

c

n

corresponding the frequencies

n

.

0.2

Derivation of Fourier integral

For seeing how the expansion (3) changes when

p

→

∞

,  we put first the expressions (4) of

c

n

to the series (3):

f

⁢

(

t

)

=

∑

n

=

-

∞

∞

e

i

⁢

n

⁢

π

⁢

t

p

⁢

1

2

⁢

p

⁢

∫

-

p

p

f

⁢

(

t

)

⁢

e

-

i

⁢

n

⁢

π

⁢

t

p

⁢

𝑑

t

By denoting

ω

n

:=

n

⁢

π

p

and

Δ

n

⁢

ω

:=

ω

n

+

1

-

ω

n

=

π

p

,  the last equation takes the form

f

⁢

(

t

)

=

1

2

⁢

π

⁢

∑

n

=

-

∞

∞

e

i

⁢

ω

n

⁢

t

⁢

Δ

n

⁢

ω

⁢

∫

-

p

p

f

⁢

(

t

)

⁢

e

-

i

⁢

ω

n

⁢

t

⁢

𝑑

t

.

It can be shown that when

p

→

∞

and thus

Δ

n

⁢

ω

→

0

,  the limiting form of this equation is

f

⁢

(

t

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

e

i

⁢

ω

⁢

t

⁢

𝑑

ω

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

e

-

i

⁢

ω

⁢

t

⁢

𝑑

t

.

(5)

Here,

f

⁢

(

t

)

has been represented as a

Fourier integral

.  It can be proved that for

validity

of the expansion (4) it suffices that the

function

f

is

piecewise continuous

on every finite interval having at most a finite amount of

extremum

points and that the

integral

∫

-

∞

∞

|

f

⁢

(

t

)

|

⁢

𝑑

t

converges

.

For better to compare to the Fourier series (3) and the coefficients (4), we can write (5) as

f

⁢

(

t

)

=

∫

-

∞

∞

c

⁢

(

ω

)

⁢

e

i

⁢

ω

⁢

t

⁢

𝑑

ω

,

(6)

where

c

⁢

(

ω

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

e

-

i

⁢

ω

⁢

t

⁢

𝑑

t

.

(7)

0.3

Fourier transform

If we denote

2

⁢

π

⁢

c

⁢

(

ω

)

as

F

⁢

(

ω

)

=

∫

-

∞

∞

e

-

i

⁢

ω

⁢

t

⁢

f

⁢

(

t

)

⁢

𝑑

t

,

(8)

then by (5),

f

⁢

(

t

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

e

i

⁢

ω

⁢

t

⁢

F

⁢

(

ω

)

⁢

𝑑

ω

.

(9)

F

⁢

(

ω

)

is called the

Fourier transform

of

f

⁢

(

t

)

.  It is an

integral transform

and (9) its

inverse

transform

.

N.B. that often one sees both the

formula

(8) and the formula (9) equipped with the same constant factor

1

2

⁢

π

in front of the integral sign.

References

1

K. Väisälä:

Laplace-muunnos

.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).

Title

Fourier series in complex form and Fourier integral

Canonical name

FourierSeriesInComplexFormAndFourierIntegral

Date of creation

2013-03-22 18:02:54

Last modified on

2013-03-22 18:02:54

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Derivation

Classification

msc 42A38

Classification

msc 42A16

Classification

msc 44A55

Related topic

FourierTransform

Related topic

KalleVaisala

Defines

Fourier integral

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