Question

Acsuming uet fution fit) aahfies the Din'chlets Conclitrons, Compute 1. I. . ue fouvier Siras 12.ue Complex Fouries seilea, &(t)t,-7sh T = 271 ,cas ut) HINT: f4)= A (An cos ut w Snwst) 11 Wo t to+T ftT As 1 scet An 1 2. Deter mi nu a Touries Anan form ethe putbench in g lt) &Ce t =0

Answer 1

T 2T o2 A - 1 Given fit) = t Fauner series expansion for fit) fIu= A + An Cosnwst+ Bn Sintnuwot F4 ahere 드 으( fI너 dt A flH Cos nuot dt T An= fiH sininwot)dt T Bn = calculahon for Fowres cocfi디ent: Ac= Ao = 2 An = t2 cos n t clt fIt cos (nuot) lt. 2TJ-T - t An = t sinnt 2t Sinint) dt An= COs nt dt 617 nt 2t cosnt (u-) -T + 2t cos nt- An= 2 cOSnt dt J-A ] tinintt Sin cnt) n2 An= nt 2tcos nt - 2 4 An= o)-( 2T COS nT- o-2n caSnT An = C An 4) cosnt 4n 4 COSNT n2 CosnT = -1) Bn-o Since fie is Eren funchon an fIe = 4 COSn t Hence + n2

Exxponenhal Fowier serien Ch eshwot ne-e here - +Ir Cn dt Jnt t2 e 27 ( 2 T 27 Cn = t nt l t 2A Jnt -Jnt 2t e -Jnt Jnt 2 e Jht 2te -Jn Gn)2 T-45)2 t2o Jnt 27 2t eJNt 2 ent -Jn +nL -(Jh)3 + 2t e ThE n2 2 e Jht 17 27 -A -JnT 1 +2Te + n2 Le 27T m 2e 27T 2.e -Jn Jn3 n2 Jn3 --gnT JhT JnT JNA 27 2 nT 27 COS nT n2 2 n3 s 2 COS nT n Jnt Hence 2 F1) PH = n2 OR 21) Int ohere fIh= COt

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