Question

a) Expand the given function in a Fourier series in the range of [-411, 41] (12 Marks) f(x) = { 1 0<x SI (sin(x) < x < 211 To what values will the Fourier Sine Series converge at x = -31, x = 0 and x = 27t? (3 Marks)

Answer 1

Solu- Let us write fex) as a fourier peries: fow= Ao + 2 (Ar essa + b sinox) 0 and evaluate the coefficients to Ay and Br .. Ao = per 5 dx + 2 r s t sinxe dx A = 2 ( x ) + 2 - Castel A r t + [-Cassit & Casti] AO - Ono I l. Casrocdict ! sinac Costa da and = .co.us Finse cortada Avui he m bere)* *m*-*3 for Ar= [shirt) + - COLET) CM (L-1)* 7 ох 2 TL 17 18

ir=1, 2, 3 ... and sinrl=0 Ar = I [-Ces (ltr) IT - Ces (17) 2017 Ar - gh L ito A = I 1-Ces (178) 29 1-Cas (18) 25 Ar = 20 L - + tr 148 Ar lt Cay and + (4 Cosz) Ar - (14 Gas emo) [i4r + Ay - ( 14 Casant) (**+1+ I (1+ Coeno) Ar= 112 Ar= -(1+ Cesaro) For=1, A = 7 / sinxcessida M s. Cine y oth A (+) [Cypx]=

How Br - 20 rxdatt sina sinrada TH (217 I Q sina sinrodos Csrx or - Car - 4 - CarIT-IL Caste- )- Gascurajare Tr TASDIT-1 sinc1-8X Sinh Casri-1 sin sinl - or the corn-1345 Fim 072 Sinutepeni orato (not) torta simons + sin emo] Br = f (GV-1)+ cinza BY (171)eta sin en x Br = H (Cert) + + sin en as T- 2 TT

osrit - Br = = + (Coson-1) to lf ofl .: Cassim = (-1) and sinrico Br = (6) nr putting the value of Ao, Ar, and Br in ① en we have esra for QIT IT (87) + + - Cha Cane costx] + [ {01943 sinna the com for Hasano) IT (8-1) Tr

Now, -(-1+1) Cinar 1-2) -(-1-1) sinx sinat git 2 Co27 Cas 40 .... 3 3185 fotovenes [ Sins + sin3x + - + ] 2 Sihoc ein 320 122 Солчо food = - 21 - 22 Case + Case +) till since + eihic sinet.nl Elence How the required Fourier series. (1924 Go + ) + ( cline snage CsYou 15

Br J 1 [1 - (18] flence 5 foco - Lisins +4+0) sim ex+ lil sinex +0+ i sin 52 +0+ sinteti] fog (2 sine sin 224 -simse fox = 1 sinxt sinax + sinhx Thus the Required pine peries is fow= + (sins + sing + Sign 52 ..)

To test for convergence foon) - 4 sinne (2ht) Now lim an = lim 11 sinne = 4X0 ho ho too D (2ht) = 0 sina opcillates blew -1 and 1} =0 (which finite) Hence the sine series is Converge at x=0, 27 and 30