find-the-fourier-series-of-the-function-f-x-x-2-x-0-x-2-0-x-2-help-me-sir-

Question Number 100966 by mhmd last updated on 29/Jun/20

f i n d t h e f o u r i e r s e r i e s o f t h e f u n c t i o n

f (x) = {\begin{cases} x - 2 ⩽ x ⩽ 0 \\ x + 2 0 ⩽ x ⩽ 2 \end{cases} h e l p m e s i r ?

Commented by bobhans last updated on 29/Jun/20

f (x) = \frac{a_{o}}{2} + \underset{n = 1}{\sum^{\infty}} b_{n} \cos (\frac{n π x}{L})

b_{n} = \frac{2}{4} [\underset{- 2}{\int^{0}} x \sin (\frac{n π x}{4}) dx + \underset{0}{\int^{2}} (x + 2) \sin (\frac{n π x}{4}) dx]

b_{n} = \frac{1}{2} [\underset{- 2}{\int^{2}} x \sin (\frac{n π x}{4}) dx + \underset{0}{\int^{2}} 2 \sin (\frac{n π x}{4}) dx]

b_{n} = \frac{1}{2} {[- \frac{4 x}{n π} \cos (\frac{n π x}{4}) + \frac{16}{{(n π)}^{2}} \sin (\frac{n π x}{4})]}_{- 2}^{2}

- \frac{1}{2} {[\frac{8}{n π} \cos (\frac{n π x}{4})]}_{0}^{2}

Commented by mhmd last updated on 29/Jun/20

h o w a r e y o u s i r ? a m d o n t a n d e r s t a n d h o w [b n c o s (\frac{n π x}{L})]

b e c a u s e t h e f o u r i e r s e r i e s i s g i v e n b y

f (x) = \frac{a o}{2} + s o m e (n ⩾ 1) [a n c o s (\frac{n π x}{L}) + b n s i n (\frac{n π x}{l})]

h o w b n c o s (\frac{n π x}{L}) a n d w h e r e a n c o s (\frac{n π x}{L}) b l e a s e s i r c a n y o u e x a c t t h i s b l e a s e

i m v e r y w a n t t h i s e q u e s t i o n ?

Commented by mathmax by abdo last updated on 29/Jun/20

sir bobhans you have supposed that f is even and this is not given in the

question you must use f (x) = \frac{a_{o}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos (nwx) + b_{n} \sin (nwx))

a_{n} = \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} f (x) \cos (nx) dx and b_{n} = \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} f (x) \sin (nx) dx (w = \frac{2 π}{T})