Calculus 3 4: chuỗi số

Definition [Half-range Fourier Series]A half range Fourier sine or cosine series is a series in which only sineterms or only cosine terms are present, respectively.. When a half range se

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

R

−L

f (x ) sinnπxL dx for all n = 1, 2, · · ·

is called the Fourier Seriesof f

The above-defined numbers an, bn are calledFourier 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)

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

1) For even function f , the function f (x ) cosnπxL is even, and function

f (x ) sinnπxL is odd, therefore,

an= 2LR0Lf (x ) cosnπxL dx ∀n = 0, 1, 2, · · · ; bn= 0 ∀n = 1, 2, · · ·

2) For odd function f , the function f (x ) cosnπxL is odd, and function

f (x ) sinnπxL is even, therefore, an= 0 ∀n = 0, 1, 2, · · ·

bn= 2LR0Lf (x ) sinnπxL dx ∀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)

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

1) For even function f , the function f (x ) cosnπxL is even, and function

f (x ) sinnπxL is odd, therefore,

an= 2LR0Lf (x ) cosnπxL dx ∀n = 0, 1, 2, · · · ; bn= 0 ∀n = 1, 2, · · ·

2) For odd function f , the function f (x ) cosnπxL is odd, and function

f (x ) sinnπxL is even, therefore, an= 0 ∀n = 0, 1, 2, · · ·

bn= 2LR0Lf (x ) sinnπxL dx ∀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)

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

1) For even function f , the function f (x ) cosnπxL is even, and function

f (x ) sinnπxL is odd, therefore,

an= 2LR0Lf (x ) cosnπxL dx ∀n = 0, 1, 2, · · · ; bn= 0 ∀n = 1, 2, · · ·

2) For odd function f , the function f (x ) cosnπxL is odd, and function

f (x ) sinnπxL is even, therefore, an= 0 ∀n = 0, 1, 2, · · ·

bn= 2LR0Lf (x ) sinnπxL dx ∀n = 1, 2, · · ·

Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 3 / 12

3) For 2L-periodic function g we haveRL

3) For 2L-periodic function g we have −LL g (x )dx = cc+2Lg (x )dx forany real constant c Therefore,

3) For 2L-periodic function g we have −LL g (x )dx = cc+2Lg (x )dx forany real constant c Therefore,

Example 2:

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Example 2:

Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 6 / 12

Example 2:

Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 6 / 12

Definition [Half-range Fourier Series]

A half range Fourier sine or cosine series is a series in which only sineterms or only cosine terms are present, respectively

When a half range series corresponding to a given function is desired, thefunction is generally defined in the interval (0, L) [which is half of the

function is specified as odd or even, so that it is clearly defined in theother half of the interval, namely, (−L, 0)

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 sineterms or only cosine terms are present, respectively

When a half range series corresponding to a given function is desired, thefunction is generally defined in the interval (0, L) [which is half of the

function is specified as odd or even, so that it is clearly defined in theother half of the interval, namely, (−L, 0)

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

function is specified as odd or even, so that it is clearly defined in theother half of the interval, namely, (−L, 0)

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 thefunction is specified as odd or even, so that it is clearly defined in theother half of the interval, namely, (−L, 0)

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 sineterms or only cosine terms are present, respectively

When a half range series corresponding to a given function is desired, thefunction is generally defined in the interval (0, L) [which is half of the

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

Half-range Fourier Cosine Series

Half-range Fourier Sine Series

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

Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 10 / 12

Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 10 / 12

3 Complex Notation: Let f satisfy Dirichlet’s conditions on (−L, L).Putting cn:= 2L1 R−LL f (x )e−inπxL dx , then the Fourier series in complexform of f is