Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

323

and

∆ α k π

1 ℓ

= (6.40) From the substitution, it can be seen that the newly introduced variable α k takes values from the interval ( 0 , + ∞ ) . Following these substitutions, the relation (6.38) becomes .

ℓ Z

∞ ∑ k = 1

1 π

∆ α k

f ( t ) cos α k ( t − x ) d t ,

f ( x )=

lim ℓ → ∞

(6.41)

− ℓ

that is, when we switch to the limit value

∞ Z 0

∞ Z

1 π

d α

f ( t ) cos α ( t − x ) d t .

f ( x )=

(6.42)

− ∞

The relation (6.42) is known in literature as the Fourier formula , and the corresponding integral as the Fourier integral .

Theorem22 Let the function f ( x ) be:

- partly smooth in every finite interval, and - absolutely integrable in interval ( − ∞ , ∞ ) .

Then the function f ( x ) can be substituted by a Fourier integral (6.42) for each x , except in points of first order discontinuity x o , in which the value of the function f ( x o ) should be substituted by f ( x o − 0 )+ f ( x o + 0 ) 2 .

The Fourier formula can also be represented by the relation f ( x )= Z ∞ 0

[ A ( λ ) cos λ x + B ( λ ) sin λ x ] d λ ,

(6.43)

where

∞ Z ∞ Z

1 π 1 π

A ( λ )=

f ( x ) cos λ x d x ,

− ∞

B ( λ )=

f ( x ) sin λ x d x .

− ∞