Question

Find the Fourier series of the following functions in the given intervals. f(x) = r +, - <x< g(t) = { inter) 0. -T<r <0, sin(x), 0<x< 1.

Answer 1

solution fox) = x+ IT -it CX < IT let the given function be expended . کفشومو نعومه شد 8=1 21 - : fto = Ao + È (Ay Coolê x)+ By xin exs] where too-1 / f sex - - - * (x+2) de = a b + 1x]" = { * +r). Co -173} = { + T* - + Ts' } = { 212} n X

And A = 2 / 2 f. fe cost 3) de = + 1 "Cx+) Curse do T ' using definite integral property 84 = x carx is odd function stre cose che =0 -TT rx =*40** (sinya)"} * { [ show – ish remo]} = - l sinst & sinnt} - sine too - Coto).

I fou sihradu LT g T Cosr T t rol L TTY2 .-#{}," nosima kada tiry "aforeta} * * {Exemple + bla Concerty = to {Ese constytahova]" - F. (es va ? fro TT Cedr TT + or E1) cos fon) } - \ {conto I cos érm) } tr{-ricornti – o coset} Cong t - Copy 1) .. hori C -2017 Comu) – įCO] = - Com OTT 18+ = -lo i Cos 17=EUM? osrt

gr) = ० . , -कट x ८० on { sime i ocact let us white foe) as a fourier series LT : Fø= A + CA, Coxx + B, please) Meta Regis" goes the ***ht ( Tarian on? = { 0 - (com x], "} =et C-lonit + cooo] - C +2.+1] :: . An of Ay = TÉ fa cose x) by IT 8=1,2,. - o. Centro de este =#+5 "stros cuerit?

J 2'sin xi Cos directe zsina CosB=SPM(A+B) + 8 inCA 6) Cotox con Ir sity Corf- - TGP ( x + xnx-y+] = # ( - campeones con myx] #1 Coperty - fitore t] *{commen- cm (up to me titra t *} = $. – CONCT +971 + 1-3 (0-1) 7 Ito . 148 1-7 os (TT - I COM CTT + 0) = -como +0 Coso Fat { thropynyt lt 40 mm = et CI+ las me { tot * - seta lt las m] ***199 = pt las me) ( ) ( + corto) T(1-82) 2 y rEl ... 1

· Ara - (+les tro) if x #1 y ITC 32-1)' for r =1 Til sinx coon.de A = Ñ . Ana II 20fmConnche = Su sf neu che A1 = (2x) - - c con 21 -.Como] A1 = (-1) A = - COD TA=0]

r f(x) sin 8x. one sr=1,2,3, ... :*** *0. storach t ' in storite} = { 0 + ]]" zon x shoxcha} sto "cas (1-12) – con 6 +82ok IsinasinB = cos A B Cod B = (sin (PX - Bin €1493x 7" to to. =* ( sho rt - simptomen (slu f=mY) - słu (a + m) ir sin arr) IT - sin (+0) IT - ४ ) sin(TT +0) = - stro I sinto) = sino 17 - sinema Cits titr) : 211 sinar ritr +1-8. 211 1?

B. = 8h To TC1-82) Gr81 To By = 0 g r&I (2 8th to 20 for Bi , 8= B = # 1 sm sinache s" sine che be te that | 28nze r. CO2A=1281ña | 2811A = 1-C002A as" -cosaz) ta #(*- sher" = *(T-0) 19 = 3]

Substituting the values of As, Ay, and by in Fourier series, we have f) = 1 + E CIA Conor cos x) + I sin a CB,F1B=B=--=0 ITC g24) 18 IT XX

XL Mal O-X r= 2 (r2 -1 . Conytt Evb fres = # + 1 Box - (0 + 47 )con ** fov) = + + Show – # ( 2023 + 2 Cest* + ---) f« =+ = = shoe - À L Sap 2x + cert ] - 20032x + 22-1. 9 Cos4x I
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