Ingenieria aeronautica

Lecture 2 Fourier Series
Abstract The aim of the Lecture is to present the statements of the theory of Fourier series. The basic questions considered below are: (i) periodic functions; (ii) Fourier series; (iii) Fourier sine series; (iv) Fourier cosine series; (v) differentiation of Fourier series.

2.1. Periodic functions Definition 2.1. (i) A function f (x) defined on R is called periodic ifthere is a number T > 0 such that f (x + T ) = f (x) for all x ∈ R. (2.1) (ii) The smallest positive T satisfying (2.1) is called the period of f (x). Example 2.1. (i) The functions cos x and sin x are periodic with period 2π, that is, cos(x + 2πn) = cos x, sin(x + 2πn) = sin x for all x ∈ R, n = 0, ±1, ±2, . . . .

(ii) Let l > 0. Obviously, for each n ∈ N = {1, 2, . . .} cos πnx = cos l sin πnx= sin l
πnx + 2πn l πnx + 2πn l

= cos πn(x+2l) , l = sin πn(x+2l) , l

so the functions cos πnx and sin πnx are periodic with period 2l. l l (iii) Let N ∈ N and let a0 , an and bn , 1 ≤ n ≤ N , are constants. The sum
N

a0 +
n=1

an cos

πnx πnx + bn sin l l

is periodic with the period 2l. (iv) Suppose that the coefficients (constant numbers) a0 , an and bn , 1 ≤ n < ∞, are suchthat the series ∞ πnx πnx a0 + an cos + bn sin l l n=1 converges for each x ∈ R. The sum of the series is the periodic function with its period 2l. .

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Example 2.2. The function f (x) = then there exists T > 0 such that

1 1+x2

is not periodic. Indeed, suppose the opposite is true,

1 1 = 2 1+x 1 + (x + T )2

for all
1 , 1+T 2

x ∈ R. that is, T = 0. The contradiction

Inparticular, this equality holds for x = 0, hence 1 = proves the statement. 2.2. Fourier series

˜ Let a function f (x) is defined on [−l, l], l > 0. Consider the 2l-periodic extension f (x) of f (x) from (−l, l] to R such that ˜ ˜ f (−l) = f (l), ˜ ˜ f (x + 2lk) = f (x), x ∈ [−l, l], k ∈ Z = {0, ±1, ±2, . . .}.

˜ Definition 2.2. The Fourier series of a 2l-periodic function f (x), x ∈ R is theexpansion of the form ∞ πnx πnx ˜(x) ∼ a0 + f an cos + bn sin , (2.2) l l n=1 where a0 , an and bn , n ∈ N are constant coefficients defined below. To define the coefficients a0 , an and bn , n ∈ N, we consider an ”ideal” in some sense case, ˜ ˜ namely, the case then the Fourier series of f (x) converges to f (x) for each x ∈ R uniformly on R: ∞ πnx πnx ˜(x) = a0 + f an cos + bn sin , (2.3) l l n=1 Integratingthe both sides in (2.3) over (−l, l) and making use of the obvious equalities
l

πnx cos dx = 0, l

l

sin
−l

πnx dx = 0, l

n ∈ N,

−l

we obtain 1 a0 = 2l

l

˜ f (x)dx.
−l

(2.4)

Multiplying (2.3) by cos πmx and sin πmx , we get l l ˜ f (x) cos πmx = a0 cos πmx + l l ˜ f (x) sin πmx = a0 sin πmx + l l
∞ n=1 ∞ n=1

an cos πnx cos πmx + bn sin πnx cos πmx , l l l lan cos
πnx l

(2.5)

sin

πmx l

+ bn sin

πnx l

sin

πmx l

.

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The direct calculation shows that
l

cos
−l l

πnx πmx cos dx = l l πmx πnx sin dx = l l
l

0, n = m, l, n = m, 0, n = m, l, n = m,

n, m ∈ N,

(2.6)

sin
−l

n, m ∈ N,

(2.7)

cos
−l

πnx πmx sin dx = 0, l l

n, m ∈ N,

(2.8)

Integrating the both equalities in (2.5) andmaking use of (2.6) - (2.8), we come to 1 an = l
l

πnx ˜ f (x) cos dx, l

1 bn = l

l

πnx ˜ f (x) sin dx, l

n ∈ N.

(2.9)

−l

−l

So finally we come to the following definition. ˜ Definition 2.3. The Fourier series of a 2l-periodic function f (x), x ∈ R is the expansion of the form ∞ πnx πnx ˜(x) ∼ a0 + f an cos + bn sin , l l n=1 where a0 , an and bn , n ∈ N are constant coefficientsdefined by (2.4) and (2.9). The coefficients a0 , an and bn , n ∈ N are called the Fourier coefficients of f(x). We remember that of points of [−l, l] so
l −l

g(x)dx does not change its value if we redefine g(x) at finite number
l l

f (x)dx =
−l l −l

˜ f (x)dx. f (x) sin πnx dx = l
l −l

f (x) cos πnx dx = l

l −l

˜ f (x) cos πnx dx, l

−l l −l

˜ f (x) sin πnx dx, l

and 1...