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Sine and Cosine Series (Sect. 10.4).

?Even, odd functions. ?Main properties of even, odd functions. ?Sine and cosine series. ?Even-periodic, odd-periodic extensions of functions.Even, odd functions.

Definition

A functionf: [-L,L]→Riseveniff for allx?[-L,L] holdsf(-x) =f(x).A functionf: [-L,L]→Risoddiff for allx?[-L,L] holdsf(-x) =-f(x).Remarks:

?The only function that is both odd and even isf= 0.?Most functions are neither odd nor even.

Even, odd functions.

Example

Show that the functionf(x) =x2is even on [-L,L].Solution:The function is even, since f(-x) = (-x)2=x2=f(x).2 f(x)f(-x) -x x xy f(x) = x?Even, odd functions.

Example

Show that the functionf(x) =x3is odd on [-L,L].Solution:The function is odd, since f(-x) = (-x)3=-x3=-f(x). f(-x) f(x) -x yf(x) = x3 x x?

Even, odd functions.

Example

(1)The functionf(x) = cos(ax) is even on [-L,L];(2)The functionf(x) = sin(ax) is odd on [-L,L];(3)The functionsf(x) =exandf(x) = (x-2)2are neither even

nor odd.-xx f(x) f(-x)f(x) = e yx x x f(x) = (x - 2) y

2?Sine and Cosine Series (Sect. 10.4).

?Even, odd functions. ?Main properties of even, odd functions. ?Sine and cosine series. ?Even-periodic, odd-periodic extensions of functions.

Main properties of even, odd functions.

Theorem

(1)A linear combination of even (odd) functions is even (odd). (2)The product of two odd functions is even. (3)The product of two even functions is even. (4)The product of an even function by an odd function is odd.

Proof:

(1)Letfandgbe even,that is,f(-x) =f(x),g(-x) =g(x).Then, for alla,b?Rholds,

(af+bg)(-x)=af(-x)+bg(-x)=af(x)+bg(x)= (af+bg)(x).Case "odd" is similar.Main properties of even, odd functions.

Theorem

(1)A linear combination of even (odd) functions is even (odd). (2)The product of two odd functions is even. (3)The product of two even functions is even. (4)The product of an even function by an odd function is odd.

Proof:

(2)Letfandgbe odd,that is,f(-x) =-f(x), g(-x) =-g(x).Then, for alla,b?Rholds, (fg)(-x)=f(-x)g(-x)= ?-f(x)??-g(x)?=f(x)g(x)= (fg)(x).Cases(3), (4)are similar.

Main properties of even, odd functions.

Theorem

If f: [-L,L]→Ris even, then?

L -Lf(x)dx= 2? L 0 f(x)dx.

If f: [-L,L]→Ris odd, then?

L -Lf(x)dx= 0.Lx-L yf(x) (-)-LL yf(x) xMain properties of even, odd functions.

Proof:

I=? L -Lf(x)dx= 0 -Lf(x)dx+? L 0 f(x)dxy=-x,dy=-dx.I=? 0 L f(-y)(-dy) +? L 0 f(x)dx= L 0 f(-y)dy+? L 0 f(x)dx.Even case:f(-y) =f(y),therefore, I=? L 0 f(y)dy+? L 0 f(x)dx?? L -Lf(x)dx= 2? L 0 f(x)dx.Odd case:f(-y) =-f(y),therefore, I=-? L 0 f(y)dy+? L 0 f(x)dx?? L -Lf(x)dx= 0.

Sine and Cosine Series (Sect. 10.4).

?Even, odd functions. ?Main properties of even, odd functions. ?Sine and cosine series. ?Even-periodic, odd-periodic extensions of functions.Sine and cosine series.

Theorem (Cosine and Sine Series)

Consider the function f: [-L,L]→Rwith Fourier expansionf(x) =a02 n=1? a ncos?nπxL +bnsin?nπxL .(1)If f is even, then b n= 0for n= 1,2,···,and the Fourier seriesf(x) =a02 n=1a ncos?nπxL ?is called aCosine Series. (2)If f is odd, then a n= 0for n= 0,1,···,and the Fourier seriesf(x) =∞? n=1b nsin?nπxL ?is called aSine Series.

Sine and cosine series.

Proof:

Iffis even, and since the Sine function is odd,then b n=1L L -Lf(x) sin?nπxL

dx= 0,since we are integrating an odd function on [-L,L].Iffis odd, and since the Cosine function is even,then

a n=1L L -Lf(x) cos?nπxL dx= 0,since we are integrating an odd function on [-L,L].Sine and Cosine Series (Sect. 10.4). ?Even, odd functions. ?Main properties of even, odd functions. ?Sine and cosine series. ?Even-periodic, odd-periodic extensions of functions. Even-periodic, odd-periodic extensions of functions. (1) Even-periodic case: A functionf: [0,L]→Rcan be extended as an even function

f: [-L,L]→Rrequiring forx?[0,L] thatf(-x) =f(x).This functionf: [-L,L]→Rcan be further extended as a

periodic functionf:R→Rrequiring forx?[-L,L] thatf(x+ 2nL) =f(x).Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the even-periodic extension off(x) =x5, with x?[0,1].Solution:y 1 1-1 f(x) = x5 xx-1

Even extension of f(x) = x5

y 1 1 5 y 1 -1 x1

Even-periodic extension of f(x) = x?

Even-periodic, odd-periodic extensions of functions. (2) Odd-periodic case: A functionf: (0,L)→Rcan be extended as an odd function

f: (-L,L)→Rrequiring forx?(0,L) thatf(-x) =-f(x),f(0) = 0.This functionf: (-L,L)→Rcan be further extended as a

periodic functionf:R→Rrequiring forx?(-L,L) andn

integer thatf(x+ 2nL) =f(x),andf(nL) = 0.Remark:Atx=±L, the extensionfmust satisfy:(a)f is odd, hencef(-L) =-f(L);

(b)f is periodic, hencef(-L) =f(-L+ 2L) =f(L). We then conclude that-f(L) =f(L),hencef(L) = 0.Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the odd-periodic extension off(x) =x5, with x?(0,1).Solution:y 1 1-1 f(x) = x5 x x 1 -1 1

Odd extension of f(x) = x5

y

Odd-periodic extension of f(x) = x

y 1 -1 1x 5? Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the even-periodic extension off(x) =x, with x?[0,1], and then find its Fourier Series.Solution:x 1 1-1y f(x) = xEven-periodic extension of f(x) = x y 1

1-1xEven-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the even-periodic extension off(x) =x, with

x?[0,1], and then find its Fourier Series.Solution:Sincefis even and periodic,then the Fourier Series is a

Cosine Series,that is,b

n= 0.From the graph:a 0= 1. a n=1L L -Lf(x)cos?nπxL dx= 2L L 0 f(x)cos?nπxL dx.a n= 2? 1 0 xcos(nπx)dx= 2 ?xsin(nπx)nπ+cos(nπx)(nπ)2? ???1 0,a n=2(nπ)2?cos(nπ)-1??a n=2(nπ)2?(-1)n-1?. Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the even-periodic extension off(x) =x, with x?[0,1], and then find its Fourier Series.Solution:Recall:b n= 0, anda n=2(nπ)2?(-1)n-1?. n= 2k?a2k=2[(2k)π]2?(-1)2k-1??a

2k= 0.n= 2k-1?a2k-1=2[-1-1][(2k-1)π]2?a

2k-1=-4[(2k-1)π]2.f(x) =12

-4π

2∞

k=11(2k-1)2cos?(2k-1)πx?.?Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the odd-periodic extension off(x) =x, with x?(0,1), and then find its Fourier Series.Solution:x 1 1-1y f(x) = x 1 x

Odd-periodic extension of f(x) = x

-1 1y Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the odd-periodic extension off(x) =x, with

x?(0,1), and then find its Fourier Series.Solution:Sincefis odd and periodic,then the Fourier Series is a

Sine Series,that is,a

n= 0. b n=1L L -Lf(x)sin?nπxL dx= 2L L 0 f(x)sin?nπxL dx.b n= 2? 1 0 xsin(nπx)dx= 2 -xcos(nπx)nπ+sin(nπx)(nπ)2? ???1 0,b n=-2nπ?cos(nπ)-0??b n=-2(-1)nnπ.Even-periodic, odd-periodic extensions of functions.

Example

Sketch the graph of the odd-periodic extension off(x) =x, with x?(0,1), and then find its Fourier Series.Solution:Recall:a n= 0, andb n=2(-1)n+1nπ.Therefore, f(x) =2π n=1(-1)(n+1)n sin(nπx).?quotesdbs_dbs14.pdfusesText_20