Question

3. Consider a function F(t) which is zero for negative t, and takes the value exp(-t/2 ) for > 0. Find its Fourier transforms, C(w) and S(w), defined in 200 F(t) = C(w) cos(wt) dw + Sw) sin(wt) do. J-00 J-00 [Hint: Use Euler's theorem.] 4. Demonstrate that Sr?)dt = 2* ["icºw) +8?(]dio, J-00 J-00 where the relation between F(t), C(w), and S(w) is defined above. This result is known as Parseval's theorem.

Answer 1

Answer MAN T o Consider the data as followed * let us assume that the equations are for the * Given collect from the data 1 * To conside the function Fit is to zero for negative I t, and its takes the value expl-t/2r) for too 1 * To find out the fouries from forms ((w), and $(w) FOH) = , Clw) cosciut) dw + S sin() sin fotdw —0 ha Now flt) = 0 7tco = atat la ف 27 ا le Y coswt dt Coswt dt Ewe things I Fey casiot dt ="/or cosit dit . اين How e (éttat cosuot) = (et le caswt) - 7. Hzc sinot We have to apply for integrals on bothsidy above equation - Jolētlir sinnt) d = Setlezsinut dt tw Sethz cos wut dt 22 T. => Etlet co wt = -1/22 I - WI and -1/2 z I, +W I sinwt . 72 ZI, o Clas) = 421 11 - strat) ->

. . _ similarly slwa Ti 5772207 - ② +3 we get the solve equations are . ((w)? + s(w) ² = 1 . un L 1 / +220) riz2zW)? - 1747²2 27 W data we have to the another considy Now - let us po PP)dt = 27(c*cm +sw7dw I ft at = rzetet e soon I{ccw #slast'] dw = '27) Te or de where willzzu and dwa lazdu du tet 70" = 2 2 L 4th 2 du = 24 I urte de finally we get the peruations are نه به امام و ردها ع 21 = ع ع م م between Att), alt), and sw) it is the relation as followed