how-to-find-the-Fourier-series-of-f-x-x-0-lt-x-lt-1-8-

Question Number 77285 by jagoll last updated on 05/Jan/20

how to find the

Fourier series of f (x) = x, 0 < x < \frac{1}{8}

Answered by john santu last updated on 05/Jan/20

because f (x) an odd function

, we used the sine series .

periode T = \frac{1}{8} and L = \frac{1}{16}

f (x) = \underset{n = 1}{\sum^{\infty}} b_{n} \sin (\frac{n π x}{L})

b_{n} = \frac{2}{(1 / 16)} \underset{0}{\int^{1 / 16}} x \sin (16 n π x) dx

= 32 {- \frac{x}{16 n π} \cos (16 n π x) + \frac{1}{{(16 n π)}^{2}} \sin (16 n π x)} \underset{0}{\overset{1 / 16}{∣}}

= \frac{32}{n π} {(- 1)}^{n}

then we get

f (x) = \underset{n = 1}{\sum^{\infty}} (\frac{32}{π n}) {(- 1)}^{n} \sin (16 n π x)

= - \frac{32}{π} \sin (16 π x) + \frac{16}{π} \sin (32 π x) - \frac{32}{3 π} \sin (48 π x) + \frac{8}{π} \sin (64 π x) - \dots

Commented by jagoll last updated on 05/Jan/20

thanks you sir

Facebook Tweet Pin