Question

A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x) = 0<x< 2 Determine the Fourier expansion of the periodic function f(x). X - TT

Answer 1

Solution Guiven funeteón flu) is a ferindie function with period it and defined as flx) = SAAT 1-TI CALO HT 2 0 <n<TT. and no oT and T and ocNcT... mx) n1 clearly, the function fem) is not defined at the parits Us -TI, 0, 77. for convenience let us take franzo at ten)= I at nao. Thus flo) being continuas an Iznes accepta at a conhere there is an ordinary discontinuity, is bounded and integrable there. Further fem) is manotonic segerenu on each of open interval - La co Thus fen) Satisfres Dirichlets condition on G71, TI]. Hence, the fourrier expansion of the funcheon fla) is gireen by flu)= ale (ancosna+by sinn where, a= # Stende, an= tffon Cosanan, brit fon seninnes Naw, 9,= * S fonda + + S Hem do - Sx - I dan = 21 [ + x3 + 21 [ 2 - x?" [- + tra] T[-T7 = + 24 * I - to -TT -U T -TT - so Mall dn & 2 + 2T Т. + -a

ना Nall cosnadnt du 2 -TT (sinnebe - IT -T an=ts cosunda + + $"*Coś mx # [wan)san me?" - to suime + + f(T) semasa ham 27 Teosna tahi to + 3+ [consumo me" [Cosnital 놂 다 T-1 = To 279 0 + 2T 11 = = 2nT 'T + 27 = 2 TT 0 2nm [-eosno] + 4i [cost [l-con+ Bosnia and bn = +1, ***T sin nada + +5,52 Tsun nx du E&+) casament cosmoa bet ga [ & Tr) eos na?" + 25 (cosnada 2 hht Ein ne or zło [5] + bani si ma?" - in in = Therefore, the fourier series expansion of fla) .+ 26.cosan-tu sinna) ce. Ifa)=-E À sinna + an /n 21 an is girien by flm) = NI mz This ee completes the solution.

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