Fourier Series. constant. The ;east value of T>0 is called the period of f(x). f(x) is well defined and single valued periodic function

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1 Fourier Series Periodic uctio A uctio is sid to hve period T i, T where T is ve costt. The ;est vlue o T> is clled the period o. Eg:- Cosider we kow tht, si si si si si... Etc > si hs the periods,,6,.. Dirichlet Coditios Here is the lest vlue The period o si Suppose tht the rbiry uctio i ii iii is well deied d sigle vlued periodic uctio hs oly iite umber o discotiuity hs iite umber o mimum d miimum These coditios re kow s dirichlet coditios.

2 Fourier series I is periodic uctio d stisctio Dirichlet Coditios, the the uctio c be epded by trigoometric series or iiite series clled Fourier series s b cos si where, d b re clled Fourier coeiciets Euler s Formule or idig Fourier coeiciets I is periodic uctio with period i the itervl c c ie, c, c d i c be represeted i trigoometric series b cos si the c c d c cos d c c db si d c The epressios givig the vlues,, & b re clled Euler s ormule. Note :- I c, the c, c e, Euler s ormule c be writte s

3 d cos d c b si d c Note :- I c -, the c,c -, Euler s ormule c be writte s d Covergece o Fourier series cos d b si d i The Fourier series o coverges to t ll poits where is cotiuous t the the sum o the Fourier series t is. ii At poit o discotiuity, the series coverges to the verge vlue o the uctio t ed poits. i,e, I is poit o discotiuity o the the vlue o the

4 Formuls F.S. t is e e si bd si b bcosb b [ ] e e c osbd cosb bsi b b [ ] uvd uv u v u v... where u, u re derivtives v, v re itegrls si A cos B 5 cos A si B 6 cos A si B 7 si A si B 8 Si 9 Cos - cos cos si si -θ -si θ cos -θ cos θ PROBLEMS: Fourier e - s Fourier series i < <. Hece deduce tht the vlue o

5 5 Further derive series or cosech sol let the Fouries series be where b cos si [ ] d e d e e e cos d e cos d e { cos si } e e cosbd [ cosb bsi b] b

6 6 e e b si d e si d e { cos si } e b e Substitutig these vlues i, we get e cos si e e cos si Deductio whe poit o cotiuity

7 7 e cos e e e > e > > e e [ e e ] sih `` `` sih `` > sih `` Q cos ech sih cos ech `` Determie the Fourier series epsio o - i <<. Deduce the sum o the series 6... sol Let the Fourier series be

8 8 ] si cos [ b d where d d d cos d d cos cos

9 9 u - u - u - v cos si v cos v si v si cos si cos o b cos d cosd si d u- u - vsi v -cos/ u - v -si/ v cos/

10 Substitute these vlues i, we get Deductio whe poit o discotiuity cos si cos b cos cos b cos cos

11 Epd - s Fouier series o period i the itervl <<.Hece deduce the sum o the series sol Let the Fourier series be where [ ] > > > si cos b d d

12 cos d u- u -- vcos v si/ u v -cos/ v -si/ si cos si cos b si d si d

13 u- u -- vsi v -cos/ u v -si/ v cos/ cos si cos b where d si d u vsi v -cos u v -si [ cos si ] [ cos ] [ ] - cos d c

14 si cos d si si d si si d u u v si si v si si cos cos v si si v cos cos si si `

15 5 whe, cos d c cos d si d si cos d si d vsi u u v -cos/ v -si/ c o s s i c o s -/ b si d s i s i d { cos cos } cos A B Cos A B { cos cos }

16 6 v cos cos u u si si v v cos cos si si ` cos cos cos cos b O whe, b si d

17 7 si si d si d cos d cos d si cos u vcos v si/ u v -cos/ b Hece the Fourier series is b b cos cos si si cos cos si

18 8 Deductio :- whe / poit o cotiuity cos cos si cos si cos I si si : < < : < < Obti the

19 9 Fourier series o periodicity sol let the Fourier series be cos si b where d d d si d cos cos cos c o s d [ ] c o s d c o s d / s i c o s d { s i s i } d c o s c o s c o s c o s c o s c o s

20 cos cos cos cos si si cos cos si si cos cos cos cos cos cos cos

21 i,e. whe, cos si cos d si d cos

22 b si d si si d si A si B cos A B cos A B [ cos cos ] d [ cos cos ] d [ Q ] si si [ ] b i whe, b si d s i s i d s i d c o s d s i [ ] b H e c e th e F o u r i e r s e r ie s i s c o s c o s s i

23 si cos cos cos Derive the Fourier epsio o cos i o< < d deduce tht Sol Let the Fourier series be b cos si where d cos d si d si d cos [ cos cos ] [ ]

24 cos cos d si cos d si cos d si si d si si cos cos cos d

25 5 b si d cos si d cos si d si si d [ Qsi Asi B cos A B cos A B] si cos d cos cos d si si si si b Hece the Fourier series is

26 6 cos cos Deductio :- Whe Poit o cotiuity cos cos >

27 7 7 Epd, <<, < < As Fourier series Sol Let the Fourier series be b cos si Where d d cos d

28 si d 8 cos b Hece the Fourier series is si Homework problems i, Fid the Fourier series o periodicity o or d hece deduce tht - i, Fid the Fourier series o period or the uctio X cos i < <. Epd e i Fourier series i,

29 9 I -, id the Fourier series o period i the itervl,. Hece deduce tht -, Epd i Fourier series o periodicity o or < <. Eve & odd uctios A uctio is sid to be eve uctio i - The grph is such uctio is symmetric bout y-is

30 Eg.,, cos, si For :- - - A uctio id sid to be dd uctio i - - The grph o such uctio is symmetric bout the origi.

31 Note : i I is eve the d d ii I is odd the d Fourier series represettio o i -, Note : I is eve the the Fourier series repesettio or i -, cotis oly costt termdcosie terms ie., cos where d cos d sie terms Note I is odd, the the Fourier series represettio or i -, cotis oly ie., b si whereb si d

32 Note I is either eve or odd the the Fourier series represettio o i -, is b cos si d cos d Problems :- b si d I i - cos si, show tht Deduce tht the sum o the series 6 Sol : Let the Fourier series be b cos si where d

33 u v cos u v si / d d d d d cos d d d d cos cos cos cos

34 u v -cos / v -si / si cos si b si d si d si d si d si d u u v si v -cos / v -si / cos si b Fourier series is cos si

35 5 Deductio : whe poit o discotiuity Fid the Fourier series o cos where is ot iteger i - < <.Deduce tht sol... cos cot

36 e 6 Give cos Here, - cos- cos > is eve Fourier series be cos where d cos d si si cos cos d [ cos cos cos d cos ] d [ cos [ cos cos ] d cos ] d si [ si cos cos ] d si si cos d

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