CHAPTER 4 FOURIER SERIES

Transcription

1 CHAPTER 4 FOURIER SERIES CONTENTS PAGE 4. Periodic Fuctio 4. Eve ad Odd Fuctio Fourier Series for Periodic Fuctio Fourier Series for Haf Rage Epasios 4.5 Approimate Sum of the Ifiite Series 9 Eercise 34 Refereces 36

2 4. Periodic Fuctio A fuctio f() t is periodic with period of T if Eampe 4.. F t T f t, T Show that f ( ) cos is a periodic fuctio ad fid its period. Soutio: Sice f ( ) cos, the f T cos T. What is the vaue of T, such that cos T cos? Note that : cos ( T ) cos cost si si T cos if This is true for T,,3,... cost si T The smaest vaue for T is, kow as fudameta period. Therefore, f ( ) cos is a periodic fuctio with period of.

3 Eampe 4.. Determie whether f si5 is a periodic fuctio. If yes, fid the period. Soutio: si5 f T T si 5 cos5t cos5 si 5T cos5t si5 if si5t cos5t cos 5T T 5 Therefore, f si5 is a periodic fuctio with the period of. 5

4 4. Eve ad Odd Fuctio f is a eve fuctio if f f f is a odd fuctio if f f, for a., for a. (I) (II) Geometricay, Eve The graph is symmetrica about the y-ais. Odd The graph is symmetrica about the origi. NOTE: If oe of (I) ad (II) are satisfied, the f is caed either eve or odd. Eampe 4.. Determie whether the foowig fuctios are eve, odd, or either. (i) f (ii) h si (iii) (iv) g 3 si f t e t 3

5 Soutios: (i) f f f. (ii) h si si( ) is a eve fuctio. si( ) (by trigoometric idetity) h Or, refer to the graph. h() -π π π. Symmetrica about the origi. - h( ) si is a odd fuctio. (iii) 3 g si( ) 3 si ( 3 si ) g ( ) 3 From (ii), si( ) si g( ) si is a odd fuctio. 4

6 (iv) f ( t) e t Argumet : Test for eve. Note that f ( t) f ( t) oy whe t. (reca defiitio: for a fuctio to be a eve fuctio, it must satisfies f ( t) f ( t) for a t ). Argumet : Test for odd. t t is ever equa to e, f ( t) f ( t). e t Argumet 3: From the graph, f t e is either symmetrica about the y-ais or about the origi. Cocusio: f t e t is either eve or odd. 5

7 4.. Properties of Eve ad Odd Fuctios Eve Eve Eve Odd Odd Odd Eve Odd Neither Eve / Eve Eve Odd / Odd Eve Odd / Eve Odd Itegra properties i) For eve fuctio: f d f d Eampe: f ( ) d d - ii) For odd fuctio: f d Eampe: f ( ) si si d si d si d. NOTE: It is importat to uderstad the itegra properties of eve ad odd fuctio before you ca proceed to the et sectio. 6

8 Eampe: If f( ) is eve, the f ( ) si d. f ( ) cos d f ( ) cos d Eve X Odd ---> Odd Eve X Eve ---> Eve If f( ) is odd, the. f ( ) si d f ( ) si d Odd X Odd ---> Eve f ( ) cos d. Odd X Eve ---> Odd 7

9 ( I ) 4.3 Fourier Series for Periodic Fuctio f( ), cosistig of Sie ad Cos. Fourier series for f( ) give by a f a cos bsi where a, a where a ad b ca be epressed as foowig: f d a f cos d b f si d T ad T is period for periodic fuctio. If f( ) is a eve fuctio, the b. Hece, its Fourier series is give by : Eve X OddOdd. Reca properties. ( II ) where a f a cos a f d a f cos d Eve properties 8

10 Reca properties If f is a odd fuctio, the a, a. Hece, its Fourier series is give by: ( III ) ad f b si b f si d Odd X Odd Eve Summary: f( ) Neither Eve Odd Formua I II III 9

11 Eampe 4.3. Fid the Fourier series for f f, f. Soutio: i) Determie T ad. T =, =. ii) Draw the graph of f( ). f() iii) Check whether f( ) is a odd or eve fuctio (or either). Idetify the correspodig formuae. From the graph, f( ) is a odd fuctio. Hece, the Fourier series formuae are give by (III):

12 iv) Cacuate the Fourier series of f(). f b si. b f si d si d si d cos si cos si ( ) cos ( ) Remarks: si cos ( ) for,,,... Therefore, ( ) f si. Note: It is ot wrog to use the fu rage of [-,] whe computig b. The resut wi be the same; it s just that you may ecouter a oger cacuatio.

13 Eampe 4.3. Fid the Fourier series for f f f Soutio: i) Determie T ad. T,. ii) Draw the graph of f( ). f() - π - π π π 3 π iii) Check whether f( ) is a odd or eve fuctio (or either). Idetify the correspodig formuae. From the graph, f( ) is a eve fuctio. Hece, the Fourier series formuae are give by (II):

14 iv) Cacuate the Fourier series of f(). a f a cos a d d ( ) Ca aso use d a f ( )cos d cos d cos d si cos si cos (cos ),,,3,... 4, odd, eve 3

15 Therefore, a ( ) a cos f where a a cos 4, odd, eve Eampe Show that the Fourier series of f ( ) si, f ( ) f ( ) 4 is give by f ( ) cos. 4 Soutio: i) Determie T ad. T,. 4

16 ii) Draw the graph of f( ). 3 3 iii) Check whether f( ) is a odd or eve fuctio (or either). Idetify the correspodig formuae. From the graph, f( ) is a eve fuctio. Hece, the Fourier series formuae are give by (II): iv) Cacuate the Fourier series of f(). a si d 4 4 cos 4 a sicos( ) d (si( ) si( )) d 5

17 cos( ) cos( ) 4 ( 4 ) 4 Hece, f ( ) cos. 4 Eampe Fid the Fourier series of, f, f f Soutio: i) Determie T ad. T,. ii) Draw the graph of f( ). f(). -3π -π π π 3π. -π 4π.. 6

18 iii) Check whether f( ) is a odd or eve fuctio (or either). Idetify the correspodig formuae. f( ) is either eve or odd fuctio. Hece, the Fourier series formuae are give by (I): iv) Cacuate the Fourier series of f(). a f a cos bsi a f d f d d d a f cos d cos d si cos si cos 7

19 cos cos,,,3,... Remarks: si si,,,3,.. cos cos,,,3,.. b si d cos si si cos si cos cos a f a cos bsi m cos m si m 4 m m m 8

20 4.4 Fourier Series for Haf Rage Epasios. Sometimes, if we oy eeds a Fourier series for a fuctio defied o the iterva [, ], it may be preferabe to use a sie or cosie Fourier series istead of a reguar Fourier series. This ca be accompished by etedig the defiitio of the fuctio i questio to the iterva [, ] so that the eteded fuctio is either eve (if we wats a cosie series) or odd (if we wats a sie series). Such Fourier series are caed haf-rage epasios. Fourier Cosie series form a f a where cos a f d a f ( )cos d,,,... Fourier Sie series form f b where si b f si d,,,3... 9

21 Eampe 4.4. (Fia Eam Sem II 4/5) Cosider the fuctio f ( ) ;. a) Sketch the appropriate graphs i the iterva -3 3 for each of the foowig cases: i) The periodic etesio of f( ). ii) The odd -periodic etesio of f( ). iii) The eve -periodic etesio of f( ). b) Fid the Fourier cosie series of f( ). Soutio: a) The origia graph of f( ) is give by f () Therefore, the soutio to a) is just a etesio to the origia graph. i) The periodic, rage f( ) 3 3

22 ii) The odd -periodic, rage Trasform the origia graph ito a odd graph. (Remember, the odd graph is symmetry about the origi.) f( ) f( ) Eted the odd -periodic graph to the rage-3 3. (Repeat the shape). f( ) period. 3 3 iii) The eve -periodic, rage Trasform the origia graph ito a eve graph. (Remember, the eve graph is symmetry about y-ais.)

23 f( ) f( ) Eted the eve -periodic graph to the rage-3 3. (Repeat the shape). f( ) 3 3 b) Fourier cosie series. a f a cos a ( ) d. a ( )cos d si cos ( ) by parts or tabuar

24 cos si cos 4 ( ), odd, eve 4 (m ) ; m,,3,.. m Therefore f (m ) 4 cos( ) ( ). m Eampe 4.4. Fid the haf rage Fourier series for f Soutio:. i) Fourier Cosie. To get the Fourier cosie series, we eed to eted f( ) to be a eve fuctio as show beow: f()

25 Hece, the Fourier cosie series of f( ) is a where f a cos, a d ad a cos d si cos cos 4, odd, eve 4 ; m,,3,.. (m ) 4 f cos m m m 4

26 ii) Fourier Sie. To get the Fourier sie series, we eed to eted f( ) to be a odd fuctio as show beow:. f() Remember! Odd fuctio Symmetry about the origi. Hece, the Fourier sie series of f( ) is give by f b si where b f si d, si d cos si si cos f ( ) si 5

27 Eampe Give that f,. f() -π π Fid the odd ad eve etesio of f( ). Soutio: i) Odd etesio Fourier sie series. To get the Fourier sie series, we eed to eted f( ) to be a odd fuctio as show beow: f() -3π -π -π.. π π 3π a a b si d si d cos cos 6

28 4, odd, eve 4, m,,3,... (m ) Therefore, 4 f si m m m ii) Eve etesio Fourier cosie series. To get the Fourier cosie series, we eed to eted f( ) to be a eve fuctio as show beow: f() -3π -π π 3π a a d cos d si a f 7

29 4.5 Approimate Sum of the Ifiite Series We ca fid the approimate sum of the ifiite series by usig Fourier series for f( ). NOTE: If A; f( ), the f () B; is ot i the iterva. A B. Eampe 4.5. If the Fourier series for is give by f,, f ( ) f ( ), f ( ) si( ), ( ) show that the approimate sum of this ifiite series is

30 Soutio: i) Epad f( ). f ( ) si( ) ( ) si si 3 si ii) Choose a appropriate vaue of. Reca: f( ) if Let, the we obtai 3 5 f si si si si si si or

31 Eampe 5.5. i) Show that ( ) 4 cos, 3 ii) By choosig a appropriate vaue of, show that ( ), ad 6 Soutio: i) Sice gover by: f ( ) is a eve fuctio, its Fourier series is a ( ) a cos f a o 3 d a cos d si cos si 3 cos 4 4 cos 3

32 Therefore, 4 f ( ) cos 3. ii) Choose a appropriate vaue of. Let. 4 cos 3 ( ) Let. 4 ( ) 3 or. 3

33 Eampe (cotiuatio of Eg. 4.4.) If the Fourier cosie series of fuctio ; f( ) ; m is give by f( ) (m ) cos( ) m, fid the sum of Soutio: Epad f( ). f 4 cos3 cos5 cos7 ( ) cos Choose a appropriate vaue of. Let. 4 f () Reca fuctio f( ). is ot i the iterva. Hece, how do we fid f ()? f ( ) ( ) () 3

34 Therefore, Eercise:. The graph of f( ) is show as beow. Fid its Fourier series. f( ) (,) (,) (,) As. 3 ( ) ( ) cos si ( ). Write the Fourier series for f ( ) sih,. As. sih ( ) si. 3. Show that if f( ) whe 3, f( ) whe 3, ad f (), the 33

35 () f( ) si Reca Eampe Usig a appropriate vaue of, fid the sum of the foowig series ( ) As Cosider the foowig graph of f( ). f( ) a) Write the fuctio of f( ). b) Determie the Fourier series of f( ). As. si(). 6. Write the Fourier Sie series for f ( ) ( ). As. si 34

36 7. Cacuate the Fourier sie series of the fuctio f ( ) o (, ). Use its Fourier represetatio to the fid the vaue of the ifiite series As. 8 si() f( ) 3 ( ) 3, the sum of series is if 3 8. Let h be a give umber i the iterva (, ). Fid the Fourier cosie series of the fuctio if h f( ) if h As. h si h f ( ) cos h END REFERENCES. Normah Maa et. a., (8), Differetia Equatios Modue, Jabata Matematik, UTM.. Nage et. a., (4), Fudametas of Differetia Equatios, 5 th ed., Addiso Wesey Logma. 3. Rue V.Churchi ad James W.B., (978), Fourier Series ad Boudary Vaue Probems, McGraw Hi. 35