Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

312

Theorem20 If f ( x ) is a periodic function, with period 2 ℓ , and it is partially smooth on the interval x ∈ [ − ℓ,ℓ ] , then the Fourier series of the function f ( x ) (6.10), the coefficients of which are defined by relations (6.12), is convergent.

The sum of this series

∞ ∑ k = 1

ℓ

k π x ℓ

k π x

a 0 2

a k cos

+ b k sin

(6.15)

s ( x )=

+

is a) s ( x o )= f ( x o ) , if f ( x ) is continuous at point x o ∈ [ − ℓ,ℓ ] , b) s ( x o )= f ( x o + 0 )+ f ( x o − 0 ) 2 c) At the ends of the interval the following is true

at points where the function f ( x ) has discontinuities.

s ( − ℓ )= s ( ℓ )= f ( − ℓ + 0 )+ f ( ℓ − 0 ) 2 . The conditions under which the series converges are known in the literature as Dirichlet 5 conditions. The proof of this theorem is not difficult, but it requires a lot of space, so we will not elaborate it here. Readers, eager for knowledge, are referred to books [12] and [38].

R Note 1. Given that, for a continuous function, f ( x − 0 )= f ( x + 0 )= f ( x ) , that is f ( x + 0 )+ f ( x − 0 ) 2 = 2 f ( x ) 2 = f ( x ) , points a) and b) can be simply replaced by

f ( x + 0 )+ f ( x − 0 ) 2 ,

s ( x )=

for all points x ∈ [ − ℓ,ℓ ] .

R Note 2. In case it is necessary to expand a function ϕ ( x ) , which is not periodic, but satisfies all other conditions from the previous Theorem, then we proceed as follows. We look for a periodic function f ( x ) , with period 2 ℓ , which can be expanded into a Fourier series, and which coincides with the initial function ϕ ( x ) in the interval ( − ℓ,ℓ ) , i.e. f ( x )= ϕ ( x ) , for x ∈ ( − ℓ,ℓ ) , while outside this interval these functions differ. This can be achieved by extending non-periodic functions, as described earlier.

6.2.1 Expanding even and odd functions into Fourier series. Fourier sine and cosine series

Recall that if the following is true for a function f ( − x )= f ( x ) for ∀ x ∈ [ − ℓ,ℓ ]

the function is even ,

f ( − x )= − f ( x ) for ∀ x ∈ [ − ℓ,ℓ ] the function is odd . 5 Peter Gustav Lejeune Dirichlet (1805-1859), German mathematician. He became famous for his important research in the fields of Fourier series and number theory.