Chapter I - tadiyosyehualashet.files.wordpress.com€¦ · Web viewExample 2. Show that the trigonometric system 1, cos nx, sin nx ... Represent each of the following functions

Unit VIII Fourier Series

Unit VIII

Fourier Series Fourier series are series of cosine and sine terms. Since sine and cosine are periodic functions

we shall study Trigonometric series representation of what we call periodic functions.

7.1 Periodic Functions; Trigonometric Series.

Definition 7.1 A function f is said to be periodic if there is some positive number p

such that:

f (x + p) = f (x) for all x dom. f.

The number p is called the period of f and the smallest number p is called the primitive

(or fundamental) period of f.

Example 1 Let f (x) = sin x.

Now dom. f = , f (x + p) = f (x) x

sin (x + p) = sin x x

sin x cos p + sin p cos x = sin x x

sin x (1 cos p) = sin p cos x x

If x = , then 1 cos p = 0.

Now 1 cos p = 0 p = 2n ,where n N.

Therefore, 2n, where n N is a period of f and 2 is the primitive period of f.

Example 2 Let g (x) = cos x.

Now dom. g = , g (x + p) = g (x) x

cos (x + p) = cos x x

cos x cos p sin p sin x = cos x x

cos x (cos p 1) = sin p sin x x

If x = , then cos p 1= 0.

Now cos p 1= 0 p = 2n, where n N.

Therefore 2n, where n N is a period of g and 2 is the primitive period of g.

Example 3. Let f (x) = tan x.

Prepared by Tekleyohannes Negussie143


Now dom. f =x x (2n + 1) , n Z.

f (x + p) = f (x) x dom. f. tan (x + p) = tan x x dom. f.

= tan x x dom. f.

tan x + tan p = tan x tan p tan 2 x x dom. f.

tan p (1 + tan 2 x) = 0 x dom. f.

tan p = 0 x dom. f.

p = n, where n N.

Therefore n, where n N is a period of f and is the primitive period of f.

Example 4. Let f (x) = sin (x), where 0.

Now dom. f = .

f (x + p) = f (x) x .

sin (x + p) = sin x x .

sin x cos p + sin p cos x = sin x x .

sin x (1 cos p) = sin p cos x x .

If x = , then 1 cos p = 0.

Now 1 cos p = 0 p = 2n, where n N.

Therefore , where n N is a period of f and is the primitive period of f.

Graphs of Periodic Functions.

The graph of a periodic function of period p is obtained by periodic representation of its

graph in any interval of length p.

Example 1. Sketch the graph of f (x) = tan x.

Solution. The primitive period of f is .



Example 2 Sketch the graph of g (x) = sin x .

Solution Dom. g. = .

Now g (x + p) = g (x) x .

sin (x + p) = sin x x .

sin (x + p) = sin x x or sin (x + p) = sin x x .

p = 2n, where n N or sin x (1 + cos p) = sin p cos x x .

p = 2n, where n N or p = (2n 1), where n N

p = n, where n N.

Therefore is the primitive period of g.

Example 3. Sketch the graph of f, where

f (x) =

and f (x + 2) = f (x) x .



Properties of Periodic Functions.

1. If f is a periodic function with period p, then for any n N np is a period of f.

Proof: (By the principle of mathematical induction)

Since f (x + n p) = f (x) it is true for n = 1.

Assume for n = k, i.e. f (x + k p) = f (x).

We need to show for n = k + 1, i.e. f (x + (k + 1) p) = f (x).

f (x + (k + 1) p) = f (x + k p + p) = f (x + p + k p) = f (x + p) = f (x).

Therefore for any n N, np is a period of f.

2. If f and g are periodic functions with period p, then for any a, b

h (x) = a f(x) + b g(x) x dom. h

is a periodic function of period p.

Proof: h (x + p) = a f(x + p) + b g(x + p)

= a f(x) + b g(x)

= h (x) x dom. h.

Therefore h is a periodic function with period p.

3. If f (x) is a periodic function with period p, then for any b and b 0



a) f (b x) is periodic with period .

b) If ( ) is periodic with period b p.

Trigonometric Series.

Definition 1.2 The system of functions:

1, cos x, sin x, cos 2x, sin 2x, cos 3x, sin 3x, …

is called a trigonometric system.

The series

a0 +

where a0, an’s and bn’s are constants is called Trigonometric series, the

constants are called coefficients.

Definition 1.3 ( Orthogonality of the Trigonometric System)

1. Let f and g be functions defined on some interval a, b. f and g are said to be

orthogonal on a, b if and only if = 0.

2. The set of functions is said to be orthogonal on a, b if and

only if = 0 for n m.

Example 1. Show that the set of functions form an

orthogonal set on , .

Solution. Let n, m N and n m.

= sin nx.

=



=

=

= 0.

Therefore form an orthogonal set on , .

Example 2. Show that the trigonometric system 1, cos nx, sin nx, … , where n N,

form an orthogonal set on , .

Solution. We need to show that:

a) = = = 0 for n, m N and m = n.

b) = = = 0 where n, m

N

and m n.

When m = n.

= = = = 0,

and = = = 0.

When m n.

=

= = 0.



=

= = 0.

=

= = 0.

Therefore 1, cos nx, sin nx, … for n N form an orthogonal set on , .

Definition 1.4

1. A function f (x) is said to be normal or normalized in a, b if and only if

= 1

The norm of a function f (x) denoted by on a, b is defined by:

=

2. An orthogonal set of functions n (x): n N is said to be an orthonormal

set on a, b if and only if

= 1 n N.

Example 1. The set of functions n (x): n (x) = sin nx, n Nis an orthonormal

set on , .

Solution. Let m, n N and m n.

=



= = 0.

Therefore n (x): n (x) = sin nx, n Nis an orthogonal set on , .

For m = n.

= =

Therefore n (x): n (x) = sin nx, n Nis an orthonormal set on , .

Exercises. 1.1

1. Show that the following functions are non-periodic functions.

i) f (x) = x

ii) g ( x) = 2x2

iii) h (x) = e x

2. Sketch the graphs of the following functions, which are assumed to be periodic of

period 2 and, for x , are given by the formulas:

1. f (x) = e x 2. f (x) = x2

3. f (x) = x

4. f (x) =

5. f (x) =

3. For the trigonometric system find the corresponding orthonormal system on , .1.2 Fourier Series of Functions of Period 2

Let us assume that f (x) is a periodic function of period 2 that can be represented by a

trigonometric series.

f (x) = a0 + (1)

i.e. the series converges and has f (x) as its sum.

Now we need to determine the coefficients a0, an and bn .

i) To determine a0 integrate (1) from to .



= +

= 2 a0

a0 =

ii) To determine an, n N. Multiply (1) by cos mx and integrate from to where

m N, fixed.

= +

If m n, then

= = = 0.

If m = n, then

= = 0.

and =

=

= +

= .

Therefore an = n N.

iii) To determine bn , n N. Multiply (1) by sin mx where m N and integrate from

to .



= +

If m n, then

= = = 0.

If m = n, then

= 0,

= = 0.

and = = .

Therefore bn = n N.

Therefore we get what we call Euler’s Formulas.

a0 =

an =

bn = , where n N.



Definition 1.5 The numbers a0, an and bn are called the Fourier coefficients of f (x).

The trigonometric series

a0 +

Where a0, an and bn are called the Fourier coefficients is called the Fourier

Series of f (x) (Regardless of convergence).

Remark: In the Euler’s Formulas the interval of integration may be replaced by any

interval of length 2.

Example 1 Find the Fourier coefficients of each of the following periodic functions.

i) f (x) = and f (x + 2) = f (x)

ii) f (x) = for 0 x 2 and f (x + 2) = f (x).

Solutions i) a0 = = +

= + = 0,

an = = +

= + = 0,

and bn = = +

= +



= ( 1 cos n)

Thus bn = .

Therefore the Fourier series of the square wave function f is:

.

ii) a0 = = = = .

an = = =

and bn = = + = .

Therefore f (x) = + .

Convergence and Sum of Fourier series

Suppose f is any function of period 2 for which the integrals

a0 =

an = n N.

and bn = n N.

exist. Then the Fourier series of f (x) is given by:

a0 + (1)

i) If the series in (1) converges to f (x), then we write:

f (x) = a0 + .



ii) If the series in (1) doesn’t converge to f (x), then we write:

f (x) a0 + .

Question. When is the Fourier series in (1) converges to f (x)?

To answer this question first we need to define piecewise continuity.

Definition 1.6 Piecewise Continuity.

A function f (x) is said to be piecewise continuous on an interval I if

i) the interval can be divided into a finite number of subintervals in each of

which f (x) is continuous.

and ii) the limits of f (x) as x approaches the end points of each subinterval are

finite.

Remark: A piecewise continuous function is one that has only a finite number of jump

discontinuities.

Examples Consider the following functions.

i) f (x) =

ii) g (x) =

f and g are examples of piecewise continuous functions.

Example. h (x) = tan x for x ( , ).

Since h (x) is discontinuous at x = and , h is

not piecewise continuous on ( , ).



Theorem 1.1 (Representation by a Fourier series) If a periodic function with period 2 is piecewise continuous in the interval

x and has a left hand derivative and a right hand derivative at each

point of that interval, then the Fourier series (1) of f (x) is convergent. Its sum

is f (x) except at a point x0 at which f (x) is discontinuous and the sum of the series

is the average of the left and the right hand limits at x0.

Example 1. Find the Fourier series of the function f where

f (x) =

and f (x + 2 ) = f (x).

Solution. The Fourier coefficients are:

a0 = an = 0 and bn = n N

Therefore the Fourier series of f (x) is

a0 + = .

Since f is piecewise continuous on ( , ) this series converges to f (x) for each

x ( , ) \ 0.

For x = 0, the Fourier series is given by

= k + k = 0

and indeed at x = 0, = 0.

Therefore f (x) = x ( , ) \ 0.

If x = , then f ( ) = k.

Hence k =



= .

Therefore

=

.

Example 2. For each of the following, find the Fourier series expansion of the function f (x)

that has period 2.

i) f (x) = x2 , x .

ii) f (x) = x sin x , 0 x 2.

iii) f (x) = x x2 , x .

Solutions. i) a0 =

an =

.

bn =

= 0.

Therefore f (x) = for x .

For x = ,

Hence 2 =

Therefore



=

ii) a0 =

an =

If n 1, then an = and if n = 1, then a1 = .

bn =

If n 1, then bn = 0 and if n = 1, then b1 =

Therefore f (x) = .

If x = , then f () = 0.

Thus = .

Therefore

= .

iii) a0 = .

an =

.

bn =



.

Therefore f (x) = + 2 x ,

If x = 0, then

= .

Therefore

= .

1.3 Functions of any Period p = 2L

Theorem 1.2 If a function f of period p = 2L has a Fourier series representation, then

the series is:

f (x) = a0 +

where the Fourier coefficients of f (x) are given by the Euler formulas:

a0 =

an = n N,

and bn = n N.

Proof. (Refer page 578 Kreyszig)

Remark: The interval of integration in the above formulas may be replaced by any

interval of length p = 2L.



Example 1. Find the Fourier series of the function f (x) with period p = 4, where

f (x) = and f (x + 4) = f (x)

Solution. p = 4 L = 2.

Now a0 = = = = 1.

an =

,

=

bn =

Therefore f (x) = 1 + .

If x = 0, then f (0) = 2.

= .

Example 2. Find the Fourier series representation of the periodic function f (x) with period

p = 1, where

f (x) = sin x for 0 x 1 and f (x + 1) = f (x).

Solution. p = 1 L = .



Thus a0 = .

an = 2

=

= +

=

=

Thus an = n N.

bn = 2

=

= = 0.

Thus bn = 0 n N.

Therefore f (x) = 2 + 4 .

If x = 0, then

.

If x = , then



1.4 Even and Odd Functions.

Definition 1.7 (Even and Odd Functions)

1. A function f (x) is even if

f ( x) = f (x) x dom. f.

2. A function g (x) is odd if

g ( x) = g (x) x dom. g.

Example 1. f (x) = cos nx , x , where n N is even.

2. g (x) = sin nx , x , where n N is odd.

Note that:

1. If f (x) is an even function, then

= 2

2. If g (x) is an odd function, then

= 0

3. The product of

i) even and odd functions is odd.

ii) two odd functions is even.

iii) two even functions is even.



Theorem 1.3 (Fourier series of Odd and Even Functions)

a. The Fourier series of an even function f of period 2L is a Fourier cosine series.

f (x) = a0 + with coefficients

a0 = and an = n N

b. The Fourier series of an odd function f of period 2L is a Fourier sine series.

f (x) = with coefficients

bn = n N.

Theorem 1.4 (Sum of Functions)

The Fourier coefficients of a sum of two function f1 and f2 are the sum of the

corresponding Fourier coefficients of f1 and f2.

The Fourier coefficients of cf are c times the corresponding Fourier

coefficients of f.

Example 1. Let f (x) = and f (x +2) = f (x) x dom. f.

Find the Fourier series of f (x).

Solution. Let f1 (x) = and f1 (x +2 ) = f1 (x) x dom. f

and let f2 (x) = k x . Then (f1 + f2) (x) = f (x) x dom. f.



But f1 (x) = and f2 (x) = k.

Therefore f (x) = k + x dom. f.

Example 2. Find the Fourier series of the function f (x) where f (x) = x + if x

and f (x + 2) = f (x) x dom. f.

Solution. Let f1 (x) = x and f2 (x) = if x and be periodic functions with period 2.

Now f1 (x) = 2 and f2 (x) = .

Therefore f (x) = + 2 x dom. f.

Example 3. Find the Fourier series representation of

f (x) = x + x2 on , .

Solution. Let f1 (x) = x and f2 (x) = x2 for x , and f1 and f2 be periodic functions with

period 2.

Now f1 (x) is odd hence it has a Fourier sine series

and bn =

= n N

Thus f1 (x) = 2 .

f2 (x) is even, hence it has a Fourier cosine series representation.

Now a0 =

and an =

= n N

Thus f2 (x) = + 4 .



Therefore f (x) = +

Note that:

If g (x) is defined x , then the function

P (x) = is even

and q (x) = is odd.

Hence g (x) = p (x) + q (x) and g ( x) = p (x) q (x).

Exercises 1.4

Represent the following functions as a sum of even and odd functions.

i) = +

ii) = +

iii) = cosh kx + sinh kx.

1.5 Half-range Expansion.

Sometimes it is required to extend a function f (x) in the range (0, L) in a Fourier series of

period 2L, and it is immaterial what the function may be outside the range 0 x L. We could

extend f (x) with period 2L and then represent the extended function by a Fourier series.

i) If we extend the function f (x) by reflecting it in the y-axis, then the Fourier series

expansion contains only the cosine terms.

ii) If we extend the function f (x) by reflecting it in the origin, then the Fourier series

expansion contains only the sine terms.



The cosine halt-range expansion is:

f (x) = a0 +

where a0 = and an = n N

The sine halt-range expansion is:

f (x) = , where bn = n

N

Example. Find the two half-range expansions of the function

f (x) = where k 0.

Solution. f (x) can be extended to (L, L) into two ways.

I. Make f (x) even on (L, L). i.e. f (x) = f ( x) x (L, L), to get Fourier cosine

expansion.

II. Make f (x) odd on (L, L). i.e. f ( x) = f (x) x (L, L), to get Fourier sine

expansion.

Fourier cosine expansion

Now a0 = = +

= + = + =

an = =



= +

=

=

Thus an = n N.

Therefore f (x) = + .

II. Fourier sine expansion.

Now an = =

= +

=

=

Thus bn = n N.

Therefore f (x) = .



Since f (x) is piecewise continuous and = = k.

= k, but

Thus = .

Therefore

= .

Exercises 1.5

1. Represent each of the following functions by a Fourier sine and cosine series extensions,

where L + .

i) f (x) = x, 0 x L.

ii) f (x) = x2, 0 x L.

iii) f (x) = sin x, 0 x L.

2. Obtain a half-range sine and cosine series for

i) ex in 0 x 1.

ii) x2 x in 0 x .


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Chapter I - tadiyosyehualashet.files.wordpress.com€¦ · Web viewExample 2. Show that the trigonometric system 1, cos nx, sin nx ... Represent each of the following functions