Question

Problem 5. (20 points) a) Given a periodic wave function of S(x) = ax -1 <x< n that has a period of 27. Determine if f(x) is an even or odd function b) Find Fourier Sine Transform of f(x)=e

Answer 1

Sol (a)! Given! f(x) = re - X be A function f is said to in domain of even if f(-x) = f(a) for all A function f is said to be odd if , fl-x) = -f(x) for all in the domain of f(-x) n(-x) - nx = -f(36) is odd function Now Thus rx an - Пх é Sol (b): To find the Fourier Sine Transform of f(x) -TK2271 Required fourier sine transform. U bn f(x) sin(aw dx NEN - - NSC = وع e Sinlux) da

Let I = - 1x Sin(nu) dx We will use -TR For that take integration by pasts to evaluate the integral as first function and sinlons as function е the second I -Mix е Sininn) da -TX 17 е sin (musdx de dx -na • Coonx dre -e n f sin mand Jax So neth Conx) hi ha fet casino) dx The cene scosnadx - les con conude I di -Ox = -le n Cosnx = -1 en Cosna -

e-rx sinnx da -1 enx Cosnx As [ 1 ex sinnx - fonet L Tix sinnx - 7² - 1x sinnxdre -1 e-T Cosna 1) Jenx I enn h2 -TX T² I е * n Cosnat TSinna nxe ] h? H Thus, I= in conx + nsin nx Els he I + t²I -11 е 11 - in cosure of 77 sinna -On ne. U I [ [ncosna & sinna h2 2 >> I = n CRO) e [nconx & sinna n² m² ha - I= ena [nconx + 7 Sinux nx ] 124712 TIX br = Thus 17 е n²+ m² [n counx + riSinox]]

= -1 n (m² 47²) e-2 Incos na + n sin nn] - e" Incoston) + n sinches) M (47²) [ eplnen" + nir + o] - 7² [nti"+0] ] neth Egh - neth Einn n (1²+1²) aunton (e-ne - (²) T (0²1²) - in Eigh (827²_1) 7 (n² + ²) er? - TIX -K x <T is Hence, fourier sine transform of f(x) = é ntv" -1) nEN. To (22 +7²) er?