Fourier sine and cosine series

Fourier sine and cosine series

One sees from the formulae

a

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

,

b

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

x

⁢

d

⁢

x

of the

coefficients

a

n

and

b

n

for the

Fourier series

f

⁢

(

x

)

=

a

0

2

+

∑

n

=

1

∞

(

a

n

⁢

cos

⁡

n

⁢

x

+

b

n

⁢

sin

⁡

n

⁢

x

)

of the

Riemann integrable

real function

f

on the

interval

[

-

π

,

π

]

,  that

•

a

n

=

2

π

⁢

∫

0

π

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

,

b

n

=

0

∀

n

if

f

is an

even function

;

•

b

n

=

2

π

⁢

∫

0

π

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

x

⁢

d

⁢

x

,

a

n

=

0

∀

n

if

f

is an

odd function

.

Thus the Fourier series of an even function mere cosine and of an odd function mere sine .  This concerns the whole interval

[

-

π

,

π

]

.  So e.g. one has on this interval

x

≡

2

(

sin

⁡

x

1

-

sin

⁡

2

⁢

x

2

+

sin

⁡

3

⁢

x

3

-

+

⋯

)

.

Remark 1

.  On the

half-interval

[

0

,

π

]

one can in any case expand each Riemann integrable

function

f

both to a cosine series and to a sine series, irrespective of how it is defined for the

negative

half-interval or is it defined here at all.

Remark 2.

On an interval

[

-

p

,

p

]

,  instead of

[

-

π

,

π

]

,  the Fourier coefficients of the series

f

⁢

(

x

)

=

a

0

2

+

∑

n

=

1

∞

(

a

n

⁢

cos

⁡

n

⁢

π

⁢

x

p

+

b

n

⁢

sin

⁡

n

⁢

π

⁢

x

p

)

have the

expressions

•

a

n

=

2

p

⁢

∫

0

p

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

π

⁢

x

p

⁢

d

⁢

x

,

b

n

=

0

∀

n

if

f

is an even function;

•

b

n

=

2

p

⁢

∫

0

p

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

π

⁢

x

p

⁢

d

⁢

x

,

a

n

=

0

∀

n

if

f

is an odd function.

Example.

Expand the

identity function

(

http://planetmath.org/IdentityMap

)

x

↦

x

to a

Fourier cosine series

on

[

0

,

π

]

.

This odd function may be replaced with the even function

f

:

x

↦

|

x

|

.  Then we get

a

0

=

2

π

⁢

∫

0

π

x

⁢

𝑑

x

=

π

and, integrating by parts,

a

n

=

2

π

∫

0

π

x

cos

n

x

d

x

=

2

π

[

/

0

π

x

sin

⁡

n

⁢

x

n

-

∫

0

π

sin

⁡

n

⁢

x

n

d

x

]

=

2

π

/

0

π

cos

⁡

n

⁢

x

n

2

=

2

π

⁢

n

2

(

(

-

1

)

n

-

1

)

)

;

this equals to

-

4

π

⁢

n

2

if

n

is an

odd integer

, but

vanishes

for each even

n

.  Thus we obtain on the interval

[

0

,

π

]

the cosine series

x

≡

π

2

-

4

π

⁢

(

cos

⁡

x

1

2

+

cos

⁡

3

⁢

x

3

2

+

cos

⁡

5

⁢

x

5

2

+

⋯

)

.

Chosing here

x

:=

0

one gets the result

π

2

8

=

1

+

1

3

2

+

1

5

2

+

…

(cf. the entry on

http://planetmath.org/node/11010

Dirichlet eta function

at 2).

Fourier

double series

.

The Fourier sine and cosine series introduced in Remark 1 on the half-interval

[

0

,

π

]

for a function of one real variable may be generalized for e.g. functions of two real variables on a rectangle

{

(

x

,

y

)

∈

ℝ

2

⁢

⋮

⁢

0

≤

x

≤

a

,

0

≤

y

≤

b

}

:

f

⁢

(

x

,

y

)

=

∑

m

=

1

∞

∑

n

=

1

∞

c

m

⁢

n

⁢

sin

⁡

m

⁢

π

⁢

x

a

⁢

sin

⁡

n

⁢

π

⁢

y

b

,

(1)

f

⁢

(

x

,

y

)

=

d

00

4

+

1

2

⁢

∑

l

=

1

∞

(

d

l

⁢

0

⁢

cos

⁡

l

⁢

π

⁢

x

a

+

d

0

⁢

l

⁢

cos

⁡

l

⁢

π

⁢

y

b

)

+

∑

m

=

1

∞

∑

n

=

1

∞

d

m

⁢

n

⁢

cos

⁡

m

⁢

π

⁢

x

a

⁢

cos

⁡

n

⁢

π

⁢

y

b

(2)

The coefficients of the

Fourier double sine series

(1) are got by the

double integral

c

m

⁢

n

=

4

a

⁢

b

⁢

∫

0

a

∫

0

b

f

⁢

(

x

,

y

)

⁢

sin

⁡

m

⁢

π

⁢

x

a

⁢

sin

⁡

n

⁢

π

⁢

y

b

⁢

d

⁢

x

⁢

d

⁢

y

where

m

=

1

,

2

,

3

,

…

and

n

=

1

,

2

,

3

,

…

The coefficients of the

Fourier double cosine series

(2) are correspondingly

d

m

⁢

n

=

4

a

⁢

b

⁢

∫

0

a

∫

0

b

f

⁢

(

x

,

y

)

⁢

cos

⁡

m

⁢

π

⁢

x

a

⁢

cos

⁡

n

⁢

π

⁢

y

b

⁢

d

⁢

x

⁢

d

⁢

y

where

m

=

0

,

1

,

2

,

…

and

n

=

0

,

1

,

2

,

…

Note.

One can use in the double series of (1) and (2) also the

diagonal summing

, e.g. for the double sine series as follows:

c

11

⁢

sin

⁡

π

⁢

x

a

⁢

sin

⁡

π

⁢

y

b

+

(

c

12

⁢

sin

⁡

π

⁢

x

a

⁢

sin

⁡

2

⁢

π

⁢

y

b

+

c

21

⁢

sin

⁡

2

⁢

π

⁢

x

a

⁢

sin

⁡

π

⁢

y

b

)

+

(

c

13

⁢

sin

⁡

π

⁢

x

a

⁢

sin

⁡

3

⁢

π

⁢

y

b

+

c

22

⁢

sin

⁡

2

⁢

π

⁢

x

a

⁢

sin

⁡

2

⁢

π

⁢

y

b

+

c

31

⁢

sin

⁡

3

⁢

π

⁢

x

a

⁢

sin

⁡

π

⁢

y

b

)

+

…

References

1

K. Väisälä:

Matematiikka IV

.  Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).

Title

Fourier sine and cosine series

Canonical

name

FourierSineAndCosineSeries

Date of creation

2013-03-22 15:42:20

Last modified on

2013-03-22 15:42:20

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Topic

Classification

msc 42A32

Classification

msc 42A20

Classification

msc 42A16

Classification

msc 26A06

Related topic

SubstitutionNotation

Related topic

IntegralsOfEvenAndOddFunctions

Related topic

CosineAtMultiplesOfStraightAngle

Related topic

ExampleOfFourierSeries

Related topic

DoubleSeries

Related topic

UniquenessOfFourierExpansion

Related topic

DeterminationOfFourierCoefficients

Related topic

TwoDimensionalFourierTransforms

Defines

Fourier sine series

Defines

Fourier cosine series

Defines

sine series

Defines

cosine series

Defines

half-interval

Defines

Fourier double sine series

Defines

Fourier double cosine series

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