Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

322

6.2.6 Fourier integral

Representation of a function by a Fourier series is widely used in many problems of mathe matical physics, but it applies only to periodic functions. We have shown how a function, defined on a finite interval ( a , b ) , can be extended to obtain an even or odd periodic unction. However, functions defined in the interval ( − ∞ , ∞ ) , which are not periodic, appear in numerous problems. This class of functions cannot be extended into periodic functions in the way described. The question arises: is it possible to extend Fourier’s idea to such functions? Observe some function f ( x ) , which is partly smooth in the interval [ − ℓ,ℓ ] . Its Fourier series (6.10) has the form f ( x )= a 0 2 + ∞ ∑ k = 1 a k cos k π ℓ x + b k sin k π ℓ x , (6.34) where coefficients a k and b k are determined by relations (6.12)

ℓ Z

ℓ Z

k π ℓ

k π ℓ

1 ℓ

1 ℓ

a k =

f ( t ) cos

t d t , b k =

f ( t ) sin

t d t , k = 0 , 1 , 2 ,...

(6.35)

− ℓ

− ℓ

Substituting (6.35) into (6.34), and given that

k π ℓ

k π ℓ

k π ℓ

k π ℓ

k π ℓ

cos

x cos

t + sin

x sin

t = cos

( x − t ) ,

we obtain

ℓ Z

ℓ Z

∞ ∑ k = 1

k π ℓ

1 2 ℓ

1 ℓ

f ( x )=

f ( t ) d t +

f ( t ) cos

( t − x ) d t .

(6.36)

− ℓ

− ℓ

The previously posed question boils down to the question: what do these integrals tend to when ℓ → ∞ ? Let us assume that the function f ( x ) is absolutely integrable in the interval ( − ℓ,ℓ ) , i.e. ℓ Z − ℓ | f ( t ) | d t < M , where M is a finite number. Using this condition, we obtain

ℓ Z

ℓ Z

1 2 ℓ

1 2 ℓ

1 2 ℓ

lim ℓ → ∞

f ( t ) d t ≤ lim ℓ → ∞

| f ( t ) | d t < lim ℓ → ∞

M = 0 .

(6.37)

− ℓ

− ℓ

Using (7.297), the relation (6.36) becomes

ℓ Z

∞ ∑ k = 1

k π ℓ

1 ℓ

f ( x )= lim ℓ → ∞

f ( t ) cos

( t − x ) d t .

(6.38)

− ℓ

Let us introduce the substitution

k π ℓ

α k =

k = 0 , 1 , 2 ,...

,

where

( k + 1 ) π ℓ

k π ℓ

π ℓ

∆ α k = α k + 1 − α k =

(6.39)

−

=