Question

2. An odd function (1) of period 27 is to be approximated by a Fourier sine series having only m terms. The error in this approximation is measured by the square of the deviation: E = L. (s1» - Etusinnes) (a) By differentiating with respect to the coefficients be find the values of by that minimize Em (b) Consider the function -(+) for -*<x<0 1*(*- *) for 0 <<< Calculate the coefficients, b, up to n = 7. Plot the series Sm = {b, sin ne for m = 1,3,5 and 7 along with f(x). Also include a plot of the residuals. (c) Calculate En form = 1,3,5, and 7 and plot Em vs m on a log-log plot. In doing so, confirm that Em decreases rapidly with increasing m. Note that for the integrals, you can use Maple, python, or Matlab.
i don't need the plot for d

literally that is all the info provided to answer the question- there is nothing missing

Answer 1

We have obtained the coefficient of the Fourier sine series.

20 Em= (1o – ben nin (my) on for minimum Em, we have, = 0 , =0,2,3,. m. d Em. . n=9, 2, 3,-.-.m minimum - = 0, don m we have. d bn estan [ S (Sos - ba mintes) an] = Now, uning didodarentiation under integral sign, we get, Da min ans(19 - Ž. b, cinks) dn 70. 3-2 4as sin ag ch + 2 } (am o fascines)on -- 3-4 tu) nin boy tu + (bv3 ]* 2.in fars) cine) sufo

For T = k , arin Gx) in kn) da = 5* Roin ra) dn. - J. [-con (@na)] da =Ra. = 27-0 Fi sin (2na) =0] 27. - ® nek, 1 2 roin (1) sin (ux) dn. for * S* [em (n=») x - coolman) a Join - [ min oya sia untuya Jak sin (n-k)x. in-n) rin ( tk) n. (tk) = 0 in and a both one integers and sin fon) =0]

Using the results ① and ② in ① we get, I f(x) nin (nn) du + bn.at = 0 n = 1,2,--, m. = = bu = £s* fcm) min (n) dhe ., m = 1,2,.-M. bn = ? 5* }(m) sin(px) dan - T timetion Since ; fx), and then [f(n) nin (m)] in win (nn) both one odd an even timetion.

② Given timetion is, f(x) = {-x(++) ( (-7) for Jo - – a nco 0 < 2 ( x . Here, f(n) ib on odd tunction with period aa. i bn = * * 5 * $6w) rin (any du: * *(x-7) sin a) on. * * [ 2012-) contra a l'ex-7)colasbe = 0+32(ex-2). cincons] to contenus)* nya.. 62 = 0, b3 = - Bon by = 0 .:6, - - bs be to, bq - 5m - Ž "sco"=1 ] air (ru)