Fourier Series: Definition, Euler's Formula, Convergence Theorem, Important Formulae

FOURIER SERIES

Periodic functions occur frequently in engineering problems. Such periodic functions are often complicated. It is therefore desirable to represent these in terms of the simple periodic functions of sine and cosine.

Definition : Fourier Series :

Here, we express a non-sinusoidal periodic function into a fundamental and its harmonics, a series of sines and cosines of an angle and its multiples of the form.

EULER'S FORMULA FOR THE FOURIER COEFFICIENTS

If a function f(x) defined in c < x < c + 2π can be expanded as the infinite trigonometric series,

Formula (1), (2) and (3) are known as the Euler formulas.

Note: Only if the constant term is taken as a0/2, formula (2) is true for n = 0.

§ Useful Integrals to establish Euler formulae :

To establish Euler formulae, the following integrals will be required."

DETERMINATION OF FOURIER COEFFICIENTS : (Euler's Formulae)

Let f (x) be represented in the interval (c, c + 2л) by the Fourier Series

We assume that the series (1) can be integrated term by term x = c to x = c + 2π

To find a0 :

Integrate both sides of equation (1) from

To find an:

Multiply both sides of (1) by cos nx and integrate from

To find bn

Multiply both sides of (1) by sin nx and integrate from

§ CHANGE OF INTERVAL

In practice, we often require to find a Fourier series for an interval which is not of length 2л.

In many problems, the period of the function to be expanded is not 2л, but some other interval say 2l.

Suppose f(x) is defined in the interval (−1, 1).

Example 2.1.1. State the Euler's formulae when f(x) is expanded as a Fourier series in c < x < c + 2 л.

Solution: The Fourier Series for f(x) in the c < x < c + 2 л

Formulas (1), (2) and (3) are known as the Euler formulas.

Example 2.1.2. Write the formula for finding Euler's constant of a Fourier series in (0, 2π).

Solution: Let the Fourier Series for f(x) in (0), 2π) be

Formulas (1), (2) and (3) are known as the Euler formulas.

Example 2.1.3. Write the formula for finding Euler's constant of a Fourier series in (-π, π ).

Solution: Let the Fourier Serics for f(x) in (− π, π) bе

Formulas (1), (2) and (3) are known as the Euler formulas.

Example 2.1.4. Write the formula for Fourier Constants for f(x) in (c, c + 2l).

Solution: The Fourier expansion for f(x) in the interval

c < x < c + 2l is given by

Example 2.1.5. Write the formula for Fourier Constants for f(x) in (0, 2l).

Solution : The Fourier expansion for f (x) in the interval 0 < x < 2l is given by

Example 2.1.6. Write the formulas for Fourier Constants for f(x) in (-l, l).

Solution: The Fourier expansion for f(x) in the interval –l < x < l is given by

CONDITIONS FOR A FOURIER EXPANSION : [Dirichlet's Conditions]

Any function f(x) can be developed as be developed as a Fourier series are constants, provided.

(i) f(x) is periodic, single-valued and finite

(ii) ƒ(x) has a finite number of finite discontinuities in any one period and has no infinite discontinuity.

(iii) f(x) has at the most a finite number of maxima and minima.

Note 1:

Dirichlet's conditions are not necessary but only sufficient for the existence of Fourier series.

Note 2 :

Peter Gustav Lejenune Dirichlet (1805 - 1859), Great German Mathematician is known for his contributions to Fourier Series and Number Theory.

Note 3 : 

tan x cannot be expanded as a Fourier series, since tanx has infinite number of infinite discontinuties, so Dirichlet's conditions are not satisfied.

Note 4 :

cosec x cannot be expanded as a Fourier series, since one of the Dirichlet's conditions is not satisfied.

Example 2.1.7 The function cannot be expanded as a Fourier series. Explain why ?

Solution :

At x = 2, f(x) becomes infinity, it has an infinite discontinuity at x = 2

So it does not satisfy one of the Dirichlet's conditions.

Hence it cannot be expanded as a Fourier series.

Example 2.1.8: Can you expand as a Fourier series in any interval.

Solution :

This function is well defined in any finite interval in the range (-∞, ∞) it has no discontinuities in the interval

Differentiate

f(x) is maximum or minimum when f'(x) = 0

(ie.,) when 4x = 0, (i.e.,) x = 0

So it has only one extreme value, and has

(i.e.,) a finite number of maxima and minima in the interval (-∞, ∞) Since it satisfies all the Dirichlet's conditions, it can be expanded in a Fourier Series in a specified interval in the range (∞, ∞∞).

Example 2.1.9: Examine whether the function sin 1/x can be expanded in a Fourier series in π ≤ x ≤ π

Solution : 

where n is zero or an integer.

For large values of n as n → ∞, the values of x as given by (2) tend to become indefinitely small and to be crowded near to the value of x = 0.

Hence, the function (1) has an infinite number of maxima and minima near x = 0, so it does not satisfy one of the Dirichlet's conditions. It cannot be expanded in a Fourier series in the range –л ≤ x ≤ л in which the point x = 0 is included.

§ Convergence Theorem on Fourier Series :

Statement:

If f(x) is a periodic function with period 2л and f(x) and f'(x) are piecewise continuous on [-л, л], then the Fourier series is convergent. The sum of the Fourier series is equal to f (x) at all points of x where f(x) is continuous. At the points of where f(x) is discontinuous, the sum of the Fourier series is the average of the right and left limits, that is 1/2 [f(x+) +ƒ (x−)]

Note :

• In the interval (0, 2л) i.e., 0 < x < 2 л

x continuous at all points except 0 and 2л

• In the interval [0, 2π] i.e., 0 ≤ x ≤ 2π

x continuous at all points including 0 and 2л

• In the interval (-л, л) i.e., -π < x < π

x continuous at all points except at -л and л

• In the interval [-л, л] i.е., -л ≤ x ≤ л

x continuous at all points including -л and л

• In the interval (-л, 0) and (0, л) i.е., -л < x < 0 and 0 < x < л

x continuous at all points excерt -л, л and 0.

Note: If f(x) is defined in the interval (0, 2л), then

Example 2.1.10 Sum the Fourier series for f (x) = 1/2 (π − x)

Solution :

Example 2.1.11: Sum the Fourier series for

Solution:

Example 2.1.12 Sum the Fourier series for

Solution :

x = 1 is a finite point of discontinuity (in the middle) of (0, 2)

Example 2.1.13: Sum the Fourier series for

Solution : 

§ IMPORTANT FORMULAE

Note :

2.1.(a) PROBLEMS UNDER THE INTERVAL (0, 2π)

Example 2.1.a(1) : If f (x) = 1/2 (π − x), find the Fourier series of period 2л in the interval (0,2 л). Hence deduce that

Solution: 

Let the required Fourier series be

Example 2.1.a(2): Expand f(x) = x(2л - x) as Fourier series in (0, 2л) and hence deduce that the sum of

Solution :

Example 2.1.a(3) : Determine the Fourier series for the function f(x) = x2 of period 2л in 0 < x < 2л.

Solution : Let the required Fourier series be

Example 2.1.a(4) : Expand f(x) = eax as a Fourier series in (0, 2л).

Solution :

Let the required Fourier series be

Example 2.1.a(5) : Obtain the Fourier series of periodicity 2л for f(x) = e-x in the interval 0 < x < 2л. Hence deduce that the value of Further derive a series for cosech л.

Solution: Let the required Fourier series be

Example 2.1.a(6) Expand in Fourier series of f(x) = x sin x for 0 < x < 2л and deduce the result

Solution: Let the required Fourier series be

Example 2.1.a(7) : Obtain the Fourier series for the function

Solution: Let the required Fourier series be

Example 2.1.a(8): If

Solution: Let the required Fourier series be

Example 2.1.a(9): Find the Fourier series of periodicity 2л for

Solution: Let the Fourier series be

Exercise 2.1.(a)

Problems under the interval (0, 2л)

1. Show that in the range 0 to 2л the Fourier series expansion for

2.1(b) PROBLEMS UNDER THE INTERVAL (0, 2l)

Example 2.1.b(1) : Find the Fourier series expansion of period 2l for the function f(x) = (l-x)2 in the range (0, 2l). Deduce that

Solution: Let the Fourier series be

Example 2.1.b(2) Find the Fourier series expansion of f (x) = ex in (0, 2l)

Solution: Let the Fourier series be

Example 2.1.b(3) Find the Fourier series of

Solution: Let the Fourier series be

Example 2.1.b(4) : Find the Fourier expansion of

Solution: 

Example 2.1.b(5) : Find the Fourier series of the function f(x)) = 2x − x2 for 0 < x < 3 and f (x + 3) = f(x).

Solution: 

Example 2.1.b(6) If

Solution: Here 2l = 2, l = 1

Let the required Fourier series be

Example 2.1.b(7) Obtain Fourier series for f(x) of period 2l and defined as follows

Solution: Let the required Fourier series be

Exercise 2.1.(b)

Problems under the interval (0, 2l)

1. Find the F.S. for the function

2. Find the F.S. io represent the following functions

3. Find the F.S. of period 21 for the function f(x) = x(2l – x) in (0, 2L). Deduce the sum of

4. Find the F.S of f (x) = x2 in (0, 2l). Hence deduce that

5. Find the F.S. expansion of ƒ (x) = x (1 − x) (2 −x) in (0, 2).

6. Find the F.S. expansion of ƒ (x) given by

7. Find the Fourier series for