5. FOURIER SERIES. Fourier Series Fourier Series Definition : A series of the form. 340 College Mathematics

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1 4 Coege Mathematis 5. FOURIER SERIES 5. Itrodutio I various egieerig probems it wi be eessary to epress a futio i a series of sies ad osies whih are periodi futios. Most of the sige vaued futios whih are used i appied mathematis a be epressed i the form. os os a + a + a + + bsi+ bsi+ withi a desired rage of vaues of. Suh a series is aed a Fourier Series i the ame of the Freh mathematiia Jaques Foureier (768-8) 5. Periodi Futios Defiitio : If at equa itervas of the absissa the vaue of eah ordiate f() repeats itsef the f() is aed a periodi futio. i.e., A futio f() is said to be a periodi futio if there eists a rea umber α suh that f( + α ) f() for a. The umber α is aed the period of f(). we have f() f( + α ) f( + α ) f( + α ).. f( + α ). E : (i) si si ( + ) si ( +4 ).... si ( + ) Hee si is a periodi futio of the period. (ii) os os( + ) os ( + 4 ).. os ( + ). Hee os is a periodi futio of the period. We defie the Fourier series i terms of these two periodi futios. Fourier Series 4 5. Fourier Series Defiitio : A series of the form a f ( ) + a os( ) + b si( ) is aed a Fourier series of f() with period i the iterva (, + ) where is ay positive rea umber ad a, a, b are give by the formuae aed Euer s Formuae : a + + f ( ) d, a f( )os( ) d + b f( )si( ) d These oeffiiets a, a, b are kow as Fourier oeffiiets. I partiuar if, the Fourier series of f() with period i the iterva (, + ) is give by a f( ) + a os + b si ad the Fourier oeffiiets are give by + a f( d ), + a f()os d + b f( )si d We sha derive the Euer s formuae for whih the foowig defiite itegras are required.

2 4 Coege Mathematis (i) (ii) (iii) (iv) (v) + d m m os d si d m os si d for a itegers m ad + + m m os os d si si d (for a itegers m ad suh that m ) m m os d si d Derivatio of Euer s Formuae a We have f ( ) + a os + b si...() To fid the oeffiiets a, a ad b, we assume that the series () a be itegrated term by term from to + To fid a, itegrate () w.r.t from to a + + b f ( ) d+ a os d a () a () b () + + si( ) d a () (usig the defiite itegras (ii) above) Fourier Series 4 + a f( d )... (a) To fid a, mutipy both sides of () by os m where m is a fied positive iteger ad itegrate w.r.t from to + + m f( )os d m + b os si d + a m m os d a os os d + a m () + a os os d+ b () [Usig the defiite itegras (ii) ad (iii) above] + m a os os d ( m ) a () + a () am () + m m am f( )os d Chagig m to we get + m m os ( ) + a d m [Usig the defiite itegras (iv) ad (v) above]

3 44 Coege Mathematis + am f( )os d (b) To fid b, mutipy both sides of () by si m where m is a fied positive iteger ad itegrate w.r.t from to + + m f( )si d a m m si d a si os d + + b m si si d a m () a () b si si d [Usig the defiite itegras (ii) ad (iii) above] + m a si si d ( m ) + + m + bm si si d ( m ) m + bm si d [Usig the defiite itegras (iv) above] bm () [usig the defiite itegra (v)] + m bm f( )si d Chagig m to we get Fourier Series 45 + b f( )si d (b) Thus the Euer s formuae (a), (b), () are proved. Cor. : I partiuar if ad, we get the Fourier series a f( ) + a os + b si where the Foureir oeffiiets are give by a f( d ), a f( )os d b f( )si d Cor. : I the above formuae if ad, we get the Fourier series a f( ) + a os + b si where the Fourier oeffiiets are give by a f ( ) d, a f( )os d b f( )si d

4 46 Coege Mathematis 5.5 Coditios for a Fourier series epasio It shoud ot be mistake that every futio a be epaded as a Fourier series. I the above formuae we have oy show that if f() is epressed as a Fourier series, the the Fourier oeffiiets are give by Euer s formua. It is very umbersome to disuss whether a futio a be epressed as a Fourier series ad to disuss the overgee of this series. However the foowig oditio aed Dirihet s oditio over a probems. a f ( ) + a os + b si provided (i) f() is bouded (ii) f() is periodi, sige vaued ad fiite (iii) f() has a fiite umber of disotiuities i ay oe period. (iv) f() has at the most a fiite umber of maima ad miima. These oditios are aed Dirihets oditios. I fat epressig a futio f() as a Fourier series depeds o the evauatio o the defiite itegras ( )os f d ad ( )si f d withi the imits to +, to or - to aordig as f() is defied for a i (, + ) (, ) or (-, ) 5.6 Iterva with as mid poit If - the the iterva (, + ) beomes (-, ) ad further if -, the iterva beomes (-, ). These itervas have as the mid poit. For futios defied i suh itervas, we osider the effet of hagig to ad assify them as eve ad odd futios. Fourier Series Eve ad odd futios A futio f() is said to be eve if f(-) f() i the give iterva (, + ) ad a futio f() is said to be odd if f(-) -f() i the give iterva (, + ) 5.7. Tests for eve ad odd mature of a futio If f() is defied by oe sige epressio, f(-) f() impies f () is eve ad f(-) -f() impies f() is odd. If f() is defied by two or more epressios o parts of the give iterva with as the mid poit, f(-) from the futio as defied o oe side of f() from the orrespodig futio as defied o the other side, impies f() is eve. f(-) from the futio as defied o oe side of -f() from the orrespodig futio as defied o the other side, impies f() is odd. Eampes : () f() + i (-, ) f(-) (-) + + f() f() is eve. () f() i (-, ) f(-) (- ) - - f() f() is odd. + i (,) () f ( ) i (, ) f ( )i (, ) ( + ) f( )i(,) f(-) -f() f() is odd 5.7. Fourier oeffiiets whe f() is eve ad odd From defiite itegras, we have

5 48 Coege Mathematis a a φ( d ) φ( d ) if φ ( ) is eve. a a ad φ( d ) if φ ( ) is odd. a (a) If f() is eve i (-, ) i.e., iff f(-) f(), the f() os is aso eve. ( ) f(-) os f() os. Sie os(-θ ) os θ ad f() si is odd. ( ) f() si f ( )si sie si(-θ ) -si θ a f( d ) f( d ) (by above defiite itegra) a f( )os d f( )os d b f( )si d f( ) fd ( ) f( )os d + I this ase if the iterva is (-, ) we get a f ( d ) a f( )os d b (b) If f() is odd i (-, ) i.e., if f( -) -f() the Fourier Series 49 f() os is aso odd i (-, ) ( ) f ( )os f ( )os ad ( )si f is eve i (-, ) ( ) f ( )si f ( )si a f( d ) a f( )os d b f ( )si d f( )si d If the iterva is (-, ) the a, a b f( )si d 5.7. Itervas with as a ed poit Itervas ike (, ) ad (, ) with as ed poit have speia features. We kow that If a a φ( d ) φ( d ) if φ( a ) φ( ) ad if φ( a ) φ( ) f( - ) f() The a f ( ) d

6 5 Coege Mathematis a f( )os d b Simiary if, i.e., if the iterva is (, ) we get a f( d ) If a f( )os d b f( ) -f() the a, a, b f( )si d Simiary If f( - ) -f() the a, a, b f( )si d WORKED EXAMPLES ) Fid the Fourier oeffiiet a for f() si i (, ) (May ) a sid [ ( os ) + os. d] [ os si ] + ) Fid the oeffiiet a for f() - i (-,) (A 999) a ( ) d Fourier Series for < < ) If for < < fid the Fourier oeffiiet a i the fourier series. L a f( )os d L L L d.os( ) d + si( ) [si( ) ] 4) Obtai the Fourier series for f() - i the iterva (-, ). (A 999) Soutio : a f( d ) ( ) d d d ( ) [ ] a

7 5 Coege Mathematis a f( )os d ( )os d os os d os d si si si b f( )si d ( )si d si si d sid os os d os si + Fourier Series 5 os ( ) os ( ) ( ) Fourier series is give by a f( ) + aos+ bsi + ( ) + + si + ( ) f ( ) + si is the required Fourier series 5) Epad f() as a Fourier series i the iterva (-, ) ad hee show that (i) + + (ii) (A 999) 6 Soutio : f() f ( ) ( ) f ( ) f ( ) is eve i (, ) b a fd ( ) d d a f( )os d

8 54 Coege Mathematis osd osd os is eve + si os si si os si + ( ) + 4( ) iea., Foureir series is a f ( ) + aos+ bsi 4( ) + os + 4( ) f ( ) + os is the required Fourier series. (i) To prove + + Put i the above Fourier series 4( ) f () + os ( ) + 4 f () ie., ie., Fourier Series (ii) To prove that Put i the Fourier series of f() 4( ) f ( ) + os 4( )., + ( ) ie 4 + ( ) + 4 ( ) 4 i.e., ) Obtai the Fourier series for f() e i (, ) Soutio : a f ( ) d ed e e e

9 56 Coege Mathematis a f( )os d eos d a a e ( aosb+ bsi b) We kow that e osbd a + b e (os+ si ) a + e os e os ) ( ) ( e e ) + ( + ) (as si si( ) ad os ( ) )) b f( )si d e si d a a e ( asib bos b) We kow that e si bd a + b e (si os ) b + e os e os( ) e ( ) e ( ) + ( + ) ( ) ( e e ) ( + ) Fourier series is a f( ) + aos+ bsi Fourier Series 57 ( )( e e ) ( e e ) + os ( + ) + ( ) ( e e ) si ( + ) e e ( )os ( ) si i.e., f( ) sih ( ) os ( ) si e as sih e 7) Obtai the Fourier series for f() i (, ) that Soutio : a f( ) d d a os d si os ( ) ad prove si os si( ) os + + ( ) ( ) +

10 58 Coege Mathematis b f( )si d si d sid si is eve os si ( ) os si ( ) ( ) ( ) Fourier series is a f ( ) + aos+ bsi ( ) si + ( ) si f ( ) is the Fourier series. Put i the Fourier series + ( ) si f + ( ) si sie si if is eve.,,5 Fourier Series 59 5 si si si ie., ) Fid the Fourier series for e - i the iterva (-, ) Soutio : a f ( ) d e d e e e e e e si h a f ( )os d e os d ( os + si ) + ( )

11 6 Coege Mathematis e ( os si ) e ( os si ) + ( ) + ( ) ( e e ) + ( ) sih ( )( e e ) + + b f( )si d e e si d ( si os ) ( ) + e ( si os ) e(si os ) + e ( ) e ( ) + + ( ) ( e e ) ( ) sih + + Fourier series is Fourier Series 6 a f ( ) + aos + bsi sih ( )sih f( ) + os + + ( )sih si + sih ( ) i.e., f ( ) [+ os + + ( ) si + 9) Epad f() si, < < i a fourier series a si d [ os + si a si.os d (si(+) -si(-) d os( + ) os( ) [ ( + ) + os( + ) os( ) ( + ) d ] + [ ( + )] + b si.si d (os( ) os( + d ) ) ]

12 6 Coege Mathematis si( ) si( + ) si( ) si( + ) ( ) d + + os( ) os( + ) + ( ) ( ) + f ( ) + os Note : Whe we derive that ) Fid the fourier series for the periodi futio f( ) i (-, ) Give f() whih is eve a The fourier series is f ( ) + a os( ) a d a f ( )os d os( ) d si( ) si( ). si d si( ) + os( ) Fourier Series 6 + os( ) (os ) (( ) ) f ( ) + (( ) )os ) Epad for < f ( ) for < as a fourier series. a d ( ) + + ( [(4 ) ( )] + + a os d ( )os d + si( ) si( ) d si si + ( ). d + os os + + ( ) + + ( )

13 64 Coege Mathematis ( ) b si d ( )si d + os os d + os os + ( ). d + os os + The fourier series is f ( ) + (( ) )os + (( ) + )os. ) Fid a fourier series for the futio < < f( ) < < a fd ( ) d d [ ] + + a f( )os( ) d osd+.os d Fourier Series 65 si si + () b f( )si( ) d ( )sid.si d + os os + [ os os + ] ( ( ) ) b is zero for, 4, 6,... 4 ad b for,, 5,.. Required fourier series 4 f ( ) +.os +.si 4 si si Note : whe 4 f ( ) ) Fid the Fourier series for os i the iterva

14 66 Coege Mathematis - < < Let f ( ) os. It is a eve futio a f ( ) + a os ; b a os d. si d os 4 a os.osd si.os d {si( ) si( )} d + + os( ) os( ) (+ ) Fourier Series 67 f 4 ( ) os 4 4) If a is ot a iteger show that for - < < sia si si si si a + + a a a Sie f() si a is a odd futio, a & a are equa to zero. b sia.sid (os( - a ) os( ad ) ) + si( a ) si( + a ) a a + os sia os si a a a os sia a os siaθ sia si a sia os.si a a ossia si si si + + a a a Eerise : I A.. Defie a Fourier series. Write the empheria formuae for the fourier oeffiiets.. Write the fourier series with period i the iterva (, + ) 4. Derive the Euer s formuae i the iterva (, + )

15 68 Coege Mathematis 5. Write the Fourier oeffiiets i the iterva (-, ) whe f() is a) eve ad b) odd. 6. Metio dirihets oditios. 7. Fid the fourier oeffiiet a for the foowig futios : - (i) f() i - < < (ii) f( ) i < < (iii) < < f ( ) < < (iv) f ( ) - < < (v) f ( ) osλ i < < (vi) f ( ) < < + < (vi) f ( ) < < 8. Fid the fourier oeffiiets a ad b for the above probems. ( marks for eah ostats) B.. Fid the Fourier series for a) f() i - < <. Hee dedue i < b) f ( ) i < < Hee dedue ) f ( ) i - < <. Hee dedue Fourier Series d) f ( ) si e) f ( ) +,, f) f ( ) os a i, a is ot a iteger. + i < g) f ( ) i < h) f ( ) ( ) i < < If f ( ) i < < i j) f ( ) ( ) i i (, ) k) f ( ) i (,) < < ) f( ) < < m) f() i < < < < ) f ( ) < < < o) f ( ) <

16 7 Coege Mathematis a < < p) f ( ) a < < + < < q) f ( ) < < r) f( ), < <, Hee II.. Show that the fourier series for f() - i (-, ) is 4 ( ) os ( Hit f() is eve ). Show that the fourier series of i < f ( ) is i < < 4 ( ) f ( ) si(+ ). Show that the fourier series of + < < f ( ) is i < < si si f ( ) si ( Hit : f() is odd ) 4. Show that the fourier series of f ( ) os i (, ) is f 4 ( ) ( os os4 ) If f( ) + for < <, show that Fourier Series ad If f() i (, ), show that f ( ) (si si + si + ) (Hit f() is odd) Aswers A. 7 (i) (ii) (iii) (iv) (v) siλ λ 4 ( ) 8. (i) a, b 4 (ii) a ( ) ; b (iii) (( ) 4 a ), b (iv) a, b (m ) ( ) λsi λ (v) a, b (vi) a, b [ ( ) ] λ 4 (vii) [ ( ) a ], b B. I 4 os os4 os6 d) f ( ) e) f ( ) os os os a f) f ( ) a a 4 os os os g) f ( )

17 7 Coege Mathematis os os h) f ( ) si 4 si si si si si si5 i) (i) f( ) os os6 os (ii) f ( ) os os os j) f ( ) ( ) k) si 4 4 ) si si si si m) si si si + si( ) ) f ( ) si() o) f ( ) + 4a si() p) f ( ) q) f ( ) + ( ( ) ) os Haf rage osie ad sie series May times, it may be required to obtai a Fourier series epasio of a futio i the iterva (, ) whih is haf the period of the Fourier series. This is ahieved by treatig (, ) as haf rage of (-, ) ad defiig f() suitaby i the other haf Fourier Series 7 i.e., i (-, ) so as to make the futio eve or odd aordig as osie series or sie series is required. a f( d ), a f( )os d ( ) a os a f + for haf rage osie series ad b f( )si d ad write the series as f ( ) b si for haf rage sie series. Simiary, i (, ) a f( d ), a f( )os d a ad f ( ) + a os b f( )si d f ( ) b si NOTE : (i) To sove a probem o Fourier series we have to fid a, a ad b ad substitute i a f ( ) + aos + bsi (ii) Fidig of a, a, b, ivoves itegratio. I most of the probems, f() osists of terms ike,,, et whih after a few differetiatio wi be zero. The geeraized formua for itegratio of the produt of two futios u ad v aed the Beroui s rue may be used for fidig a ad b. 4 uvd uv uv + uv u v + where dashes deote differetiatio w.r.t ad suffies,,,... deote itegratio w.r.t.

18 74 Coege Mathematis os si os For eg. sid + (iii) The foowig vaues of osie ad sie are usefu os, os (-) os (-), os if is odd ad os ( ) if is eve. si, si si (), si ( ) if is odd ad si is eve. (iv) Itegratio work a be redued to a great etet by usig the ideas of eve ad odd futios, wheever is the mid poit. (v) If f() is either odd or eve, the f() may osist of some terms whih whe take idividuay may be odd or eve ad the itegratio work a be redued. Worked Eampes : ) Fid the haf rage sie series for f() i (, ) (May ) f ( ) L b si L where b f( )si d L L b.si d os + os d os si + ( ) Fourier Series 75 os ( ) Haf ramge Sie series is ( ).si( ) ) Obtai the haf -rage Sie series for f() over the iterva (, ) (A ) b.si d os ( ) os d + os si + os ( ) Haf ramge Sie series is ( ) f ( ).si. ) Fid the haf rage Fourier sie series of f() i the iterva (, ) (N ) b si d os ( ) + os( ).d os si si +. d os os + +

19 76 Coege Mathematis os os + ( ) ( ) f( ) + si( ) 4) Fid the haf rage osie series for the futio f() i (, ) (A ) It is required to fid a f ( ) + a os L where L a f( d ) ; a f( )os d L L L a a d os( ) d si( ) si( ). d os( ) os( ) +.d os( ) si( ) + ( ) 4( ) 4( ) f ( ) + os( ) () L Fourier Series 77 4( ) os( ) + 5) Fid the haf rage Sie series for f( ) i the itera << ( )si b d ( )( ) ( )( ) ( )( ) os si os + 4 ( os ) 8 8 f ( ) si 8 si si5 si ) Fid haf rage sie series of < f ( ) < < b f( )si d si d ( )si d + os si

20 78 Coege Mathematis os si + ( ) ( ) os( ) si( ) os( ) si si 4 si si5 f ( ) si ) Fid the haf rage sie series for f() - i the iterva (, ) (A ) b ( )si( d ) os si( ) ( )( ) ()( os f ( ) ( os )si( ) + 8. Fid the haf rage osie series for the futio of f() ( - ) i the iterva < <. ( ) a ( ) d + ( ).os( ) a d Fourier Series 79 ( ) os( d ) si os si ( ) ( ) + si( ) os() f ( ) + os 9) Epad f() as a osie haf rage series i < < Soutio : The graph of f() is a straight ie. Let us eted the futio f() i the iterva (-, ) so that the ew futio is symmetri a about they y ais ad hee it represets a eve futio i (-, ) the Fourier oeffiiet b a f ( ) + a os 4 a d d a os d si os 4 si os +

21 8 Coege Mathematis os os (os ) [( ) ] a f ( ) + a os 4 [( ) + ]os 5 4 os os os f ( ) os os os.., ( ) ie f Importat Note : It must be eary uderstood that we epad a futio i < < as a series of sies ad osies merey ookig upo it as a odd or eve futio of period. It hardy matters whether the futio is odd or eve. ) Epad f ( ) if < < 4 if < < 4 i the Fourier series of sie terms Soutio : Let f() be a odd futio i (-, ) a ad a ad b f ( )si d f( )si d Fourier Series 8 sid+ si d 4 4 os si ( ) 4 os si + ( ) 4 si os + os+ 4 4 si os os si.., [ ( ) ] ie b sie os 4 b ; b b 4 ; b 4 + b5 4 ; b 6 et. 5 5 f ( ) b si 4 4 si si si

22 8 Coege Mathematis ) Fid the sie ad osie series of the futio f() - i < <. (A 99) Soutio : (i) Fourier sie series: b f ( )si d ( )si d os si ( ) ( ) os si ( ) ( ) + Fourier sie series is f ( ) b si si si+ si + si + (ii) Fourier osie series: a f( ) d ( d ). a ( )os d Fourier Series 8 si os ( ) ( ) os os. os ( os ) [ ( ) )] a f ( ) + a os [ ( ) + ]os + os os os os os os ) Fid the Fourier series epasio with period to represet the futio f() i the rage (, ) Soutio : We have ad + a f( d ) ( ) d

23 84 Coege Mathematis [9 9] + a f( )os d ( )os d si os ( ) () si ++ ( ) si os si b f( )si d + ( )si d Fourier Series 85 os si os ( ) ( ) ( ) os ( 4)9 (7) si os os a f ( ) + aos + bsi 8 + os + si 8 f ( ) os + si ) If f ( ), show that os f ( ) + i the rage of (, ) Soutio : It is a eve futio b a f ( ) d d 4 ( ) [( ) ]

24 86 Coege Mathematis ( ) 6 a f( )os d os d si ( ) os si + + The Fourier series is a f ( ) + aos+ bsi os + + os f ( ) + 4) Fid the fourier Series epasio of osh a i(-, ) Soutio : f() osh a a osh ad sih a osh ad a sih a a a oshaos d Fourier Series 87 a a e + e os d a a e osd e os d + a aos+ si a asiaos e e + a + a + a aos+ si a asiaos e e + a + a + aos si si aos e + e a + a + a a e a e a( ) ( ) + a + a + a + a + a( ) a a ( e e ) a + a( ) ie.., a sih a ( a + ) b \ The Fourier series for osh a is a osha + a os a( ) sih a siha os a + ( a + ) Eerise. Fid the haf rage Fourier osie series for f() i <

25 88 Coege Mathematis < <. Prove that f ( ) 4 4 < <, 4si the sie series is [ ( ) ] si. Fid the haf rage Fourier sie series for f() e i the iterva (, ) 4. Fid the haf rage osie series for a < < f ( ) a a < < a, 5. Obtai a haf rage osie series for f() for < <. Hee show that Fid a Fourier sie series for < < a) f ( ) < < b) f ( ) ( ) i < < 7) Epad f(), - < < i a fourier series. (N ) 8) Obtai the Fourier series for f ( ) e i (, ) (N ) 9) Obtai the Fourier series for f() i (-, ) (N ) a ) Obtai the Fourier series for f ( ) e i (, ) Fourier Series 89 ( ) ad hee dedue that os eh (A ) + ) Prove that i < < 4 5 os os os ad dedue that Aswers 4 4 (i) (ii) 4 4 ( ) 96 9 ) f ( ) ( ) os ) f ( ) ( ) si + a 8 6 4) f ( ) os os os a 6 a a 6) a) f ( ) ( os )si 8 si() b) f ( ) ( ) 9) 4 os os a sia asih a ) e + ( ) os a ( a + ) asih a + ( ) si ( a + ) EXERCISE A. Defie Haf rage a) osie b) sie series

26 9 Coege Mathematis. Fid the osie ad sie series for f() i ad hee show that 8 5. Obtai the Fourier series for the periodi futio f() defied for < < by f ( ) + for < < ad hee show that Obtai the Fourier series of f() defied by + i < f ( ) i < 4. Prove that the Fourier series epasio of ( - ) defied i os os4 os6 the iterva (, ) is Obtai the Fourier series for the futio for < < f( ) for < i < < 6. f ( ) i < < 4 si si si5 Show that (i) f ( ) os os6 os (ii) f ( ) Fourier Series 9 i 7. If f ( ) ( ) i i the iterva (, ) fid the Fourier series of f() i (, ) 8. If f ( ) fid the Fourier series i (,) i (-, ) 9. Fid the haf-rage osie series for si i (, ). Fid the haf rage sie series for f() i (, ). Fid the haf rage osie series for f() o (, ). Fid the haf rage sie series for f() i (, ). Fid the Fourier series for f() + + i (-, ) 4. Epress f() + as a Fourier series i (, ) i (,) 5. Epad f ( ) as a Fourier series i (, ) i (-, ) ( ) ( ) [( ) ] + os+ si 6 for < < 6. If f ( ) si for < < si os Prove that + ad hee show that 4 ( ) for <, f 4 7. If f ( ) for < prove that os os5 f ( ) os + + ad hee show that 4 5

27 9 Coege Mathematis for 8. f ( ) for 8 os os os5 Prove that f ( ) For f( ) i (, ), prove that 4 os os5 f ( ) os For f ( ) si i (, ) fid the Fourier series ad hee dedue that ( ) Prove that the Haf rage fourier sie series for f() - i (, ) is si [ Marks]. Prove that the Haf rage sie series for f() e i (, ) is [ ( ) e]si + ANSWERS 4 () i os os os ( ii) si si+ si+ + 4 os os os Fourier Series 9 4 os os os os os os si 4 si4 si si os os os ( ) 8. si 4 os 9. 4 si si si ( ) 4os ( ) [( ) ]4 + si + ( )4 ( ) + os + si [( ) ]4 [( )( + ) ] si os os+ os os Fiite Sie ad Cosie Trasforms Defiitios: If f() is a setioay otiuous futio over some fiite iterva (, ) of the variabe, the the fiite Fourier Sie ad Cosie Trasforms of f() over (, )are defied by

28 94 Coege Mathematis F( ) Fd where,,, s s ad Fs ( ) f( )os d where,, I the iterva (, ) we have F ( ) ( )os s f d where,, ad F ( ) ( )os f d,, Usig Fourier Sie ad Cosie haf-rage series, the iverse trasforms i the iterva (, ) are give by f ( ) Fs ( ) si ad f ( ) F() + F( )os where F () ( ) f d I the iterva (, ), the above resut beomes F( ) Fs ( )si where F ( ) F () + F ( )os F () ( ) f d NOTE : If the iterva is ot give i the probems, the we have to take the iterva as (, A). WORKED EXAMPLES : Fourier Series 95 () Fid the fiite Fourier sie ad osie trasforms of f() i (, ) Soutio : Give : f(), i (, ) (, ) () We kow Fs ( ) f ( ) si d si d [usig ()] Cos os ( ) Fs ( ) Aso, F () fcos ( ) d Cos d [usig ()] () Si F() if,,,... If the F () [ ] Cosd [usig ()] () Fid the fiite Fourier sie ad Cosie trasforms of f() i (, ). Soutio : We kow Fs ( ) Fsi ( ) d

29 96 Coege Mathematis si d (usig give data) Usig Berouie s rue, we get os si Fs ( ) os si + + [ ] + ( ) Fs ( ) where,,,... Now F ( ) f( )os d [ os ] [ si si] os d (usig give data) Usig Berouie s rue, we get si os F ( ) si os + Fourier Series 97 ( ) (os os) + F ( ) [( ) ] where,,,... If, F () d F () () For the futio f(), fid the fiite Fourier sie ad Cosie trasforms i (, ) Soutio Give : f(), (, ) (, ) () We kow Fs ( ) f( )si d Usig Berouie s rue, we get si d [usig ()] ( ) os si Fs [ os ] ( si si ) [ os ] + ( ) Fs ( ) where,,,... Aso,

30 98 Coege Mathematis F ( ) f( )os d os d [usaig ()] si os ( ) [ os F ] F ( ) (os os) F ( ) {( ) } If, 4,6,..., F () If,, 5,...., ( ) F If, F () d (usig Berouie s rue) F () (4) Fid the fiite Fourier sie trasform of f() i (, ) Soutio Give : f(), (, ) (, ) () We kow Fs ( ) f ( ) si d si d [usig ()] Usig Berouie s rue, Fourier Series 99 os si os F ( ) ( ) () s os + os ( si si ) 6 (4os ) + (os os) Fs ( ) ( ) + [( ) ] (5) Fid the fiite Fourier sie ad Cosie trasforms of f() -. Soutio : Sie the rage is ot give we sha take the iterva as (, ) We kow Fs ( ) ( )si d( f( ) ad ) Usig Berouie s rue, F ( ) s Aso ( ) ( ) os ( ) si Fs [ ( )os ] ( si si ) [ ( ) ] F ( ) ( )os d Usig Berouie s rue,

31 4 Coege Mathematis ( ) ( ) si ( ) os Fs [ os ] ( si si ) [ os os ] F ( ) ( ) where. Whe, F () ( ) d F () (6) Fid the fiite Fourier Sie ad Cosie trasforms of f() - Soutio :Sie the rage is ot give, we sha take the iterva as (, ) Give : f( ),(, ) (, ) () We kow Fs ( ) f( )si d ( )si d [usig ()] Usig Berouie s rue, os si os Fs ( ) ( ) ( ) + ( ) Fourier Series 4 ( )os [ os ] ( )os [os ] ( )( ) [( ) ] + ( ) ( ) Fs ( ) [( ) ] where,,,.. Aso F ( ) f( )os d ( )os d Usig Berouie s rue, si os si F ( ) ( ) ( ) + ( ) ( )os ( )os If, F () [ ] ( )os os where [ ] () ( ) F d

32 4 Coege Mathematis 7) Show that the fiite Fourier sie trasform of f() (-) is 4 if is odd ad if is eve. Soutio : Sie the rage is ot give, we sha take the iterva as (, ) Give : f() (-), (, ) (, ) () We kow Fs ( ) f( )si d ( )si d [usig ()] ( )si d Usig Berouie s rue, os si os F ( ) ( ) ( ) + ( ) s ( )os [os ]( si si ) [ ] [os os] [( ) ] Fs ( ) [ ( ) ] If is odd, Fs ( ) [ ( )] 4 F ( ) s If is eve, Fs ( ) [ ] Fourier Series 4 4 Thus, F ( ) s if is odd ad if is eve. (8) Show that the fiite Fourier Cosie trasform of, if,,, f ( ) is, if. Soutio : We sha take the iterva as (, ) Give : f ( ),(, ) (, ) We kow Fs ( ) f( )os d () os d [usig ()] Usig Berouie s rue, si os si ( ) ( ) ( ) F ( ) + os ( si si ) [ os] F ( ) for If, F () d

33 44 Coege Mathematis [ ] F () (9) Fid the Fourier Cosie trasform of f() defied by, < < f ( ), < < Soutio : Give (, ) (, ) We kow F ( ) f ( )os d f( )os d f ( )os + f( )os d.os d+ ( )os d () si si (usig give data) Fourier Series 45 si si ( ) si F for () Whe, Thus, F (). d+ ( ) d [usig ()] [ ] [ ] F () (), for,,4,6, F ( ) ( ) ( ), for,,5, [Usig () & ()] () Fid the fiite Fourier Cosie trasform of the futio, for < f ( ), for < < Soutio : Give : (, ) (, ) We kow F ( ) f( )os d f( )os d

34 F 46 Coege Mathematis f ( )os + f( )os d.osd+ os d (usig give data) os d () si si si si, if iseve ( ) ( ) ( ), if isodd If F() osd [usig ()].d F () Thus,, for F ( ),,4,6, ( ) ( ), for,,5, Fourier Series 47 () Fid the fiite Fourier Cosie ad Sie trasforms of the futio f() e a i (, ). Soutio : We kow F ( ) f( )os d a F ( ) e os d () a a e ( aosb+ bsi b) Usig e osbd we get a + b F ( ) e a aos + si a + + a a a a a e os e si + + a a a e os e si + a + a + a a F ( ) [( ) e ] where,,,... a + whe, we get a F( ) e d [usig ()] a e a a e F () a Aso, Fs ( ) f( )si d

35 48 Coege Mathematis a Fs ( ) e si d a a e ( asib bos b) Usig e sibd, we get a + b asi os a F ( ) e a + a a a e si e os + + a a a e os a + a Fs ( ) [ ( ) e ] where,,,... a + () Fid f() i (, ) give that the fiite Fourier Cosie os( ) trasform is F ( ) ( + ) Soutio : I the iterva, we kow f ( ) f() + f( )os Here f ( ) f () + f ( )os () os( ) Give : f ( ) ( + ) f () Usig these i (), we get Fourier Series 49 os( ) f ( ) + os ( + ) () Fid f() i (, ) give that the fiite Fourier sie trasform os is fs ( ) Soutio : We kow fs( ) fs( )si i (, ) Here f( ) Fs ( )si ( os ) f ( ) si (Usig give data) os ( ) if is eve ad if is odd. Usig this, we get f ( ) si,5, 4 si si si5 f ( ) () () (5) (4) Fid f() i < < 4 Give that F ()6, f ( ) ( ) where,,,... Soutio : We kow f ( ) f() + f( )os i < < Give : 4 f ( ) (6) + [( ) ]os 4 4 (usig give data)

36 4 Coege Mathematis f ( ) 4+ os,,5, 4 5 f( ) 4 os os os [NOTE : S [f()] deote the fiite Fourier sie trasform of f() ad S - is its iverse. Simiary C [f()] deote the fiite Fourier Cosie trasform of f() ad C - is its iverse.] os (5) Show that S ( ) os Soutio : We sha prove that S ( ) Here S ( ) ( )si d ( )si d Usig Berouie s rue, S ( ) os si os ( ) ( ) + ( ) os os + ( )os Fourier Series 4 Cos S ( ) os S ( ) ksi k (6) Show that C os k( ) k where k. ksi k Soutio : To prove that C [os k( )] k Here. We kow [os k( )] os k( )os d [ o s( k k+ ) + os( k k )] d [ o s{ k ( k )}] d + [ o s{ k ( k+ )}] d si[ k ( k ) ] si[ k ( k+ ) ] + ( k) ( k + ) [si si k] + [si( ) si k] ( k ) ( k+ ) sik si k + k k + ksi k C [os k( )] k ksi k C os k( ) k

37 4 Coege Mathematis 5. Fiite Sie ad Cosie Trasforms of Derivatives. I the iterva (, ), we prove the foowig resuts. ( r) ( r) () Fs f ( ) F f ( ) ( r) ( r) ( r) ( r) () F s f ( ) ( ) f () f () + F s f ( ) Proof : Fourier fiite sie trasform is give by ( r) ( r) F s f ( ) f ( )si d Usig itegratio by parts, ( r ) ( r) ( r) Fs f ( ) f ( )si f ( )os d ( r ) ( r ) ( r ) f ()si f ()si f ( )os d ( r) ( r) F s f ( ) F f ( ) () ( r) ( r) Aso, F f ( ) f ( )os d Usig itegratio by parts, ( r) ( r) ( r) F f ( ) f ()os + f ( )si d ( r) ( r ) ( r) f ()os f ()os + F s f ( ) F f f f F f ( r) ( r) ( r) ( r) ( ) ( ) () () + s ( ) () [NOTE : Usig the above resuts () ad (), we obtai the foowig resuts i the iterva (, )] Usig r i () ad (), we get Fs F ( ) F[ f ( ) ] () Fourier Series 4 F f ( ) [( ) f () f ()] + Fs[ f ( ) ] (4) Usig r i () ad (), we get Fs[ f ( ) ] F[ f ( ) ] Fs[ f ( ) ] [( ) f () f()] F [ ( )] s f (5) [Usig (4)] Aso, F[ f ( ) ] [( ) f () f ()] + Fs[ f ( )] F[ f ( ) ] ( ) f () f () F[ ( )] f (6)[usig ()] I the iterva (, ), the above resuts beomes F f ( ) F[ f( )] (7) s [ ] [ ] s [ ] [ ] F f ( ) [( ) F( ) f()] + F[ f( )] (8) F f ( ) [( ) f( ) f ()] F[ f( )] (9) s F f ( ) [( ) f ( ) f ()] F[ f( )] () s WORKED EXAMPLES (7) By empoyig the fiite Fourier Cosie trasform, sove the equatio Y + Y e, Y () Y ( ). Soutio : Give : Y + Y e Usig fiite Fourier Cosie trasform, we get, F[ Y ] + F[ Y] F[ e ] () I the iterva, (, ), we have F[ f ( ) ] ( ) f () f () F[ ( )] f Here (, ) (, ) ad Y f() F y ( ) y ( ) y () F( y) [ ] s

38 44 Coege Mathematis Give : Y () Y ( ) [ ] [ ] F y F y () Aso, F[ e ] e os d e ( os+ si ) + e os + e os + F[ e ] ( ) e + for () Usig () ad () i (), we get F[ y] + F[ y] ( ) e + ( ) F [ y] ( ) e + + [( ) e ] F[ y] ( + )( ) This is deoted by f () for f () + ( ) e ( + )( ) (4) Put i (4) e e f () (5) Usig iverse Fourier Cosie trasform, y f () + f ( )os Fourier Series 45 ( ) e y ( e ) + os ( + )( ) (8) Empoyig the fiite Fourier sie trasform, sove the differetia equatio y + y i, give y() y(). Soutio : y + y Usig fiite Fourier sie trasform, we get Fs[ y ] + Fs[ y] Fs[ ] () I (, ), we get Fs[ f ( ) ] [( ) f () f ()] F[ ( )] s f Fs[ y ] [( ) y () y()] F[ ] s Y [ Y f( ) ] Usig Y() Y ( ), we get Fs[ y ] F [ ] s y () Aso, F s si d Usig Berouies rue, os si os + Fs os os + os [ os ] + + ( ) Fs [ ] + ( ) ()

39 46 Coege Mathematis Usig () ad () i (), we get, ( ) + F[ ] [ ] ( ) s y + Fs y + ( ) + [ ] ( ) Fs y + + ( ) ( ) Fs [ y] + Usig iverse fiite Fourier sie trasform, we get ( )si y Fs y 4 + ( ) ( ) y + si (9) Usig the fiite Fourier Sie trasform, sove the differetia equatio y + ky i < < give that y() y( ) ad k is a o-itegra ostat. Soutio : Give : y + ky Usig fiite Fourier sie trasform, F[ y ] + kf [ y] F[ ] () s s s I (, ) Fs[ y ] [( ) y( ) y()] Fs[ y] Usig y() y( ), we get F[ y ] F[ y] () s Aso, s [ ] si s F d Usig Berouie s rue, os si os si F [ s ] Fourier Series 47 [ ] os 6 os Fs + + os 6 os 6 os 6 [ ] ( ) Fs () Usig () ad () i () we get (6 ) [ ] [ ] ( ) Fs y + kfs y ( ) (6 ) Fs [ y] (4) ( k ) Usig iverse fiite Fourier sie trasform, we get y Fs ( y)si ( )(6 ) ( k ) y si

40 48 Coege Mathematis EXERCISES. Fid the fiite Fourier Sie trasforms of the foowig (a) i (, ) (b) - i (, ) () a i (, a) (d) os (e) e -. Fid the fiite Fourier Cosie trasforms of the foowig (a) i (, ) (b) ( - ) i (, ) () i (, a) a. Fid the Fourier Cosie trasform of the futio i < < f ( ) i < < 4. Fid the Fourier Cosie trasform of the futio i < < f ( ) i < < 5. Show that the Fiite Fourier Sie trasform of is ( ) + a 6. Show that the fiite Fourier Sie trasform of f ( ) e i + a (, ) is [ + ( ) e ] a + 7. Fid the fiite Fourier Sie trasform of (i) si a ad (ii) os a. 8. Fid the fiite Fourier Cosie trasform of si a. 9. Fid f() i (, ) give. os (a) Fs( ) Fourier Series 49 (b) Fs ( ) os () F( ),,,,..., F () (d) F( ),,,,..., F (). If k is a ostat ad < <, the prove that k osh k ( k) C k + sih a. Sove the foowig differetia equatios. (a) y y e,, give y () y ( ) usig Fourier fiite Cosie trasform. (b) y y si i give y() y( ) usig Fourier fiite Sie trasform. () y y e i, give y() y( ) usig Fourier fiite Sie trasform. (d) y + y si i, give y() y( ) usig Fourier fiite Sie trasform. (e) y + y si, < < give y () y ( ) usig Fourier fiite Cosie trasform. ANSWERS ( ) + 4. (a) (b) { ( )}a () (d) for ad + { ( ) } for,, 4,. (e) [ ( ) e ] +

41 4 Coege Mathematis. ( ) ( ) (a) (b) [( ) ] a ( ). + ( ) os 4. si 7. (i) if aais, aiteger ad,,,... Fs ( ) i aisapositive, it eger. [ + ( ) os a ] (ii) F ( ) a if a, iseve 8. F ( ) a if a, isodd a 9. (a) os si (b) si os () + si (d) + si e 4 ( ) e. (a) y + os 4 ( + )( + 4) Fourier Series 4 (d) y + si,,5,... 4 (e) 4 y ( ) ( 4 + ) os (b) 4 y si+ si 8 ( ),4,6,.. [( ) e ] () y si ( + )