Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

310

and

∞ ∑ k = 1

ℓ

k π x ℓ

k π x

a 0 2

f ( x )=

a k cos

+ b k sin

(6.10)

+

We have introduced here the following substitutions a 0 2

= A 0 , a k = A k sin ϕ k , b k = A k cos ϕ k , 2 ℓ = T .

Functions f N and f defined in this way are periodical with period 2 ℓ . The series (6.10) is called a trigonometric series . When observing the series (6.10) the following question arises: is it possible to represent a function f ( x ) by a trigonometric series? The aim of this chapter is to give an answer to this question. Let us first establish the relation between coefficients a n and b n and the function f ( x ) itself. If the series converges uniformly then it can be integrated element-by-element, which yields: ℓ Z − ℓ f ( x ) d x = a 0 ℓ + ∞ ∑ k = 1   a k ℓ Z − ℓ cos k π x ℓ + b k ℓ Z − ℓ sin k π x ℓ   ⇒ ⇒ a 0 2 = 1 2 ℓ ℓ Z − ℓ f ( x ) d x . (6.11)

m π x ℓ

m π x ℓ

Multiplying the relation (6.10) by cos

and sin

, respectively, and then integrating

from − ℓ to ℓ , we obtain:

ℓ Z

ℓ Z

m π x ℓ

m π x ℓ

1 ℓ

1 ℓ

a m =

f ( x ) cos

d x ; b m =

f ( x ) sin

d x .

(6.12)

− ℓ

− ℓ

Thus, we have seen how, starting from a trigonometric series, the relation between a function and its corresponding expansion into a series can be established, that is, the relation between the series coefficients and the function. Coefficients a k and b k established in this way are called Euler 3 coefficients of the Fourier 4 series of function f ( x ) . However, in this way we have not determined whether the Fourier series of the function f ( x ) , with coefficients a k and b k determined in this way, converges to the function f ( x ) . Thus, to each integrable function f ( x ) , on the interval [ − ℓ,ℓ ] , we can relate a trigonometric series f ( x ) ∼ a 0 2 + ∞ ∑ k = 1 a k cos k π x ℓ + b k sin k π x ℓ , (6.13) where coefficients a k and b k are determined by expressions (6.12). The symbol ” ∼ ” is used in this case to indicate that the convergence of the observed series has not yet been established. When its convergence (that is, conditions the function f ( x ) must satisfy) is proved, then the symbol ” ∼ ” can be replaced by "=". 3 Leonhard Euler (1707-1783), famous Swiss mathematician. He contributed to almost all areas of mathematics and its applications in problems of physics. Special emphasis should be placed on his contribution in the field of differential and difference equations, Fourier series, special functions, complex analysis, calculus of variations, mechanics and hydrodynamics. 4 Jean-Baptiste Joseph Fourier (1768-1830), French physicist and mathematician. He set the foundations of Fourier series in his principal work Théorie analytique de la chaleur .