Infinite Series and Differential Equations

Nguyen Thieu Huy

Hanoi University of Science and Technology

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

1 / 12

Fourier Series

Let f (x) be defined on R, periodic with period 2L, and piecewise

continuous on (−L, L).

∞

a0

nπx

1 Is there any series of the form

+

an cos nπx

such

L + bn sin L

2

n=1

∞

a0

nπx

an cos nπx

+

∀x ∈ (−L, L) ?

that f (x) =

L + bn sin L

2

n=1

2 Suppose such a series as above exists, is it unique?

Definition

Let f (x) be as above. Then, the series of trigonometric functions

L

∞

a0

nπx

1

+

b

sin

in

which

a

=

+

an cos nπx

f (x) cos nπx

n

n

L

L

L

L dx for

2

n=1

−L

all n = 0, 1, 2, · · · ; bn =

1

L

L

−L

f (x) sin nπx

L dx for all n = 1, 2, · · ·

is called the Fourier Series of f .

The above-defined numbers an , bn are called Fourier Coefficients of f .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 12

Dirichlet’s Theorem

Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L).

Suppose that f (x) exists and is piecewise continuous on (−L, L). Then

the Fourier series of f is convergent for all x ∈ R to the following sum

∞

a0

nπx

=

an cos nπx

+

L + bn sin L

2

n=1

f (x)

if f is continuous at x,

= 1

if f isn’t continuous at x,

2 (f (x + 0) + f (x − 0))

where an and bn are Fourier coefficients defined as above.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 12

Dirichlet’s Theorem

Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L).

Suppose that f (x) exists and is piecewise continuous on (−L, L). Then

the Fourier series of f is convergent for all x ∈ R to the following sum

∞

a0

nπx

=

an cos nπx

+

L + bn sin L

2

n=1

f (x)

if f is continuous at x,

= 1

if f isn’t continuous at x,

2 (f (x + 0) + f (x − 0))

where an and bn are Fourier coefficients defined as above.

Remark.

1) For even function f , the function f (x) cos nπx

L is even, and function

nπx

f (x) sin L is odd, therefore,

L

an = L2 0 f (x) cos nπx

L dx ∀n = 0, 1, 2, · · · ; bn = 0 ∀n = 1, 2, · · · .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 12

Dirichlet’s Theorem

Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L).

Suppose that f (x) exists and is piecewise continuous on (−L, L). Then

the Fourier series of f is convergent for all x ∈ R to the following sum

∞

a0

nπx

=

an cos nπx

+

L + bn sin L

2

n=1

f (x)

if f is continuous at x,

= 1

if f isn’t continuous at x,

2 (f (x + 0) + f (x − 0))

where an and bn are Fourier coefficients defined as above.

Remark.

1) For even function f , the function f (x) cos nπx

L is even, and function

nπx

f (x) sin L is odd, therefore,

L

an = L2 0 f (x) cos nπx

L dx ∀n = 0, 1, 2, · · · ; bn = 0 ∀n = 1, 2, · · · .

2) For odd function f , the function f (x) cos nπx

L is odd, and function

f (x) sin nπx

is

even,

therefore,

a

=

0

∀n

=

0, 1, 2, · · ·

n

L

2 L

nπx

bn = L 0 f (x) sin L dx ∀n = 1, 2, · · ·

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 12

L

3) For 2L-periodic function g we have −L g (x)dx =

any real constant c. Therefore,

c+2L

an = L1 c

f (x) cos nπx

L dx ∀n = 0, 1, 2, · · ·

1 c+2L

nπx

bn = L c

f (x) sin L dx ∀n = 1, 2, · · ·

for any real constant c.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

c+2L

g (x)dx

c

for

4 / 12

L

3) For 2L-periodic function g we have −L g (x)dx =

any real constant c. Therefore,

c+2L

an = L1 c

f (x) cos nπx

L dx ∀n = 0, 1, 2, · · ·

1 c+2L

nπx

bn = L c

f (x) sin L dx ∀n = 1, 2, · · ·

for any real constant c.

Example.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

c+2L

g (x)dx

c

for

4 / 12

L

3) For 2L-periodic function g we have −L g (x)dx =

any real constant c. Therefore,

c+2L

an = L1 c

f (x) cos nπx

L dx ∀n = 0, 1, 2, · · ·

1 c+2L

nπx

bn = L c

f (x) sin L dx ∀n = 1, 2, · · ·

for any real constant c.

Example.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

c+2L

g (x)dx

c

for

4 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 12

Example 2:

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sine

terms or only cosine terms are present, respectively.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sine

terms or only cosine terms are present, respectively.

When a half range series corresponding to a given function is desired, the

function is generally defined in the interval (0, L) [which is half of the

interval (−L, L), thus accounting for the name half range].

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sine

terms or only cosine terms are present, respectively.

When a half range series corresponding to a given function is desired, the

function is generally defined in the interval (0, L) [which is half of the

interval (−L, L), thus accounting for the name half range]. Then the

function is specified as odd or even, so that it is clearly defined in the

other half of the interval, namely, (−L, 0).

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Half-range Fourier Sine Series

Let f be defined on (0, L) and satisfying Dirichlet’s conditions. Extend f

to an odd function f˜ on (−L, L) by putting

f (x)

if x ∈ (0, L),

f˜(x) :=

−f (−x) if x ∈ (−L, 0).

Outside (−L, L), f˜ is periodically extended to R. Then, Fourier coefficients

of f˜ (hence of f ) are an = 0 ∀n = 0, 1, 2, · · · ;

L˜

L

f (x) sin nπx dx = 2

f (x) sin nπx dx ∀n = 1, 2, · · · .

bn = 2

L

0

Nguyen Thieu Huy (HUST)

L

L

0

L

Infinite Series and Diff. Eq.

9 / 12

Half-range Fourier Sine Series

Let f be defined on (0, L) and satisfying Dirichlet’s conditions. Extend f

to an odd function f˜ on (−L, L) by putting

f (x)

if x ∈ (0, L),

f˜(x) :=

−f (−x) if x ∈ (−L, 0).

Outside (−L, L), f˜ is periodically extended to R. Then, Fourier coefficients

of f˜ (hence of f ) are an = 0 ∀n = 0, 1, 2, · · · ;

L˜

L

f (x) sin nπx dx = 2

f (x) sin nπx dx ∀n = 1, 2, · · · .

bn = 2

L

0

L

L

0

L

Half-range Fourier Cosine Series

Let f be as above. Extend f to an even function f˜ on (−L, L) by putting

f (x)

if x ∈ (0, L),

f˜(x) :=

f (−x) if x ∈ (−L, 0).

Outside (−L, L), f˜ is periodically extended to R. Then, Fourier coefficients

L˜

L

of f˜ (hence of f ) are an = 2

f (x) sin nπx dx = 2

f (x) cos nπx dx

L

0

L

∀n = 0, 1, 2, · · · ; bn = 0 ∀n = 1, 2, · · · .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

L

0

L

9 / 12

Example

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

10 / 12

Example

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

10 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

11 / 12

Nguyen Thieu Huy

Hanoi University of Science and Technology

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

1 / 12

Fourier Series

Let f (x) be defined on R, periodic with period 2L, and piecewise

continuous on (−L, L).

∞

a0

nπx

1 Is there any series of the form

+

an cos nπx

such

L + bn sin L

2

n=1

∞

a0

nπx

an cos nπx

+

∀x ∈ (−L, L) ?

that f (x) =

L + bn sin L

2

n=1

2 Suppose such a series as above exists, is it unique?

Definition

Let f (x) be as above. Then, the series of trigonometric functions

L

∞

a0

nπx

1

+

b

sin

in

which

a

=

+

an cos nπx

f (x) cos nπx

n

n

L

L

L

L dx for

2

n=1

−L

all n = 0, 1, 2, · · · ; bn =

1

L

L

−L

f (x) sin nπx

L dx for all n = 1, 2, · · ·

is called the Fourier Series of f .

The above-defined numbers an , bn are called Fourier Coefficients of f .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 12

Dirichlet’s Theorem

Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L).

Suppose that f (x) exists and is piecewise continuous on (−L, L). Then

the Fourier series of f is convergent for all x ∈ R to the following sum

∞

a0

nπx

=

an cos nπx

+

L + bn sin L

2

n=1

f (x)

if f is continuous at x,

= 1

if f isn’t continuous at x,

2 (f (x + 0) + f (x − 0))

where an and bn are Fourier coefficients defined as above.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 12

Dirichlet’s Theorem

Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L).

Suppose that f (x) exists and is piecewise continuous on (−L, L). Then

the Fourier series of f is convergent for all x ∈ R to the following sum

∞

a0

nπx

=

an cos nπx

+

L + bn sin L

2

n=1

f (x)

if f is continuous at x,

= 1

if f isn’t continuous at x,

2 (f (x + 0) + f (x − 0))

where an and bn are Fourier coefficients defined as above.

Remark.

1) For even function f , the function f (x) cos nπx

L is even, and function

nπx

f (x) sin L is odd, therefore,

L

an = L2 0 f (x) cos nπx

L dx ∀n = 0, 1, 2, · · · ; bn = 0 ∀n = 1, 2, · · · .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 12

Dirichlet’s Theorem

Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L).

Suppose that f (x) exists and is piecewise continuous on (−L, L). Then

the Fourier series of f is convergent for all x ∈ R to the following sum

∞

a0

nπx

=

an cos nπx

+

L + bn sin L

2

n=1

f (x)

if f is continuous at x,

= 1

if f isn’t continuous at x,

2 (f (x + 0) + f (x − 0))

where an and bn are Fourier coefficients defined as above.

Remark.

1) For even function f , the function f (x) cos nπx

L is even, and function

nπx

f (x) sin L is odd, therefore,

L

an = L2 0 f (x) cos nπx

L dx ∀n = 0, 1, 2, · · · ; bn = 0 ∀n = 1, 2, · · · .

2) For odd function f , the function f (x) cos nπx

L is odd, and function

f (x) sin nπx

is

even,

therefore,

a

=

0

∀n

=

0, 1, 2, · · ·

n

L

2 L

nπx

bn = L 0 f (x) sin L dx ∀n = 1, 2, · · ·

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 12

L

3) For 2L-periodic function g we have −L g (x)dx =

any real constant c. Therefore,

c+2L

an = L1 c

f (x) cos nπx

L dx ∀n = 0, 1, 2, · · ·

1 c+2L

nπx

bn = L c

f (x) sin L dx ∀n = 1, 2, · · ·

for any real constant c.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

c+2L

g (x)dx

c

for

4 / 12

L

3) For 2L-periodic function g we have −L g (x)dx =

any real constant c. Therefore,

c+2L

an = L1 c

f (x) cos nπx

L dx ∀n = 0, 1, 2, · · ·

1 c+2L

nπx

bn = L c

f (x) sin L dx ∀n = 1, 2, · · ·

for any real constant c.

Example.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

c+2L

g (x)dx

c

for

4 / 12

L

3) For 2L-periodic function g we have −L g (x)dx =

any real constant c. Therefore,

c+2L

an = L1 c

f (x) cos nπx

L dx ∀n = 0, 1, 2, · · ·

1 c+2L

nπx

bn = L c

f (x) sin L dx ∀n = 1, 2, · · ·

for any real constant c.

Example.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

c+2L

g (x)dx

c

for

4 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 12

Example 2:

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sine

terms or only cosine terms are present, respectively.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sine

terms or only cosine terms are present, respectively.

When a half range series corresponding to a given function is desired, the

function is generally defined in the interval (0, L) [which is half of the

interval (−L, L), thus accounting for the name half range].

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sine

terms or only cosine terms are present, respectively.

When a half range series corresponding to a given function is desired, the

function is generally defined in the interval (0, L) [which is half of the

interval (−L, L), thus accounting for the name half range]. Then the

function is specified as odd or even, so that it is clearly defined in the

other half of the interval, namely, (−L, 0).

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 12

Half-range Fourier Sine Series

Let f be defined on (0, L) and satisfying Dirichlet’s conditions. Extend f

to an odd function f˜ on (−L, L) by putting

f (x)

if x ∈ (0, L),

f˜(x) :=

−f (−x) if x ∈ (−L, 0).

Outside (−L, L), f˜ is periodically extended to R. Then, Fourier coefficients

of f˜ (hence of f ) are an = 0 ∀n = 0, 1, 2, · · · ;

L˜

L

f (x) sin nπx dx = 2

f (x) sin nπx dx ∀n = 1, 2, · · · .

bn = 2

L

0

Nguyen Thieu Huy (HUST)

L

L

0

L

Infinite Series and Diff. Eq.

9 / 12

Half-range Fourier Sine Series

Let f be defined on (0, L) and satisfying Dirichlet’s conditions. Extend f

to an odd function f˜ on (−L, L) by putting

f (x)

if x ∈ (0, L),

f˜(x) :=

−f (−x) if x ∈ (−L, 0).

Outside (−L, L), f˜ is periodically extended to R. Then, Fourier coefficients

of f˜ (hence of f ) are an = 0 ∀n = 0, 1, 2, · · · ;

L˜

L

f (x) sin nπx dx = 2

f (x) sin nπx dx ∀n = 1, 2, · · · .

bn = 2

L

0

L

L

0

L

Half-range Fourier Cosine Series

Let f be as above. Extend f to an even function f˜ on (−L, L) by putting

f (x)

if x ∈ (0, L),

f˜(x) :=

f (−x) if x ∈ (−L, 0).

Outside (−L, L), f˜ is periodically extended to R. Then, Fourier coefficients

L˜

L

of f˜ (hence of f ) are an = 2

f (x) sin nπx dx = 2

f (x) cos nπx dx

L

0

L

∀n = 0, 1, 2, · · · ; bn = 0 ∀n = 1, 2, · · · .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

L

0

L

9 / 12

Example

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

10 / 12

Example

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

10 / 12

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

11 / 12

## Chuong 4. Chuoi so - chuoi luy thua.ppt

## dai so 11 CB tuan 3+4

## so theo doi chung cu khoi lop 3,4,5

## Bộ đề ôn thi vào ĐHCD đề số 1,2,3,4,5.doc

## Các số 1.2.3.4.5

## Đại số 9 tiết 3,4

## KTTT HK I (Unit 1,2,3,4) + KEY (SO HOT)

## Đại số 7tiết 3-4

## De anh 9 so 3 + 4 ( 2010 - 2011)

## 2 đề kiểm tra Học kỳ I Môn Toán lớp 11 tham khảo và đáp án số 3 +4

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