Let f(x) = x for-4 sx<4. Because f is odd on [-4, 4), its Fourier cosine coefficients are all zero. Its Fourier sine coefficients are b, =L* sin ("). dx =i * sin ("*) dz=(-1)**128"m² -6 n' The Fourier series of x' on [-4,4] is sin n

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Odd function
Let f(x) = x for -4<x<4. Because f is odd on [-4, 4], its Fourier cosine coefficients are
all zero. Its Fourier sine coefficients are
b, =L
' sin
dx
)dx= (-1)*+*128*m² – 6
-
2, 'sin
The Fourier series of x' on [-4, 4] is
00
E(-1)*+128"² -6
sin
We will make use of this discussion later, so here is a summary of its conclusions:
If f is even on (-L, L), then its Fourier series on this interval is
1
za0+Ea, cos
in which
2
-fx) cos () dx for n=0,1,2,....
If f is odd on [-L, L], then its Fourier series on this interval is
Eb, sin (")
where
2
f(x) sin (") dx for n=1,2,..
Transcribed Image Text:Odd function Let f(x) = x for -4<x<4. Because f is odd on [-4, 4], its Fourier cosine coefficients are all zero. Its Fourier sine coefficients are b, =L ' sin dx )dx= (-1)*+*128*m² – 6 - 2, 'sin The Fourier series of x' on [-4, 4] is 00 E(-1)*+128"² -6 sin We will make use of this discussion later, so here is a summary of its conclusions: If f is even on (-L, L), then its Fourier series on this interval is 1 za0+Ea, cos in which 2 -fx) cos () dx for n=0,1,2,.... If f is odd on [-L, L], then its Fourier series on this interval is Eb, sin (") where 2 f(x) sin (") dx for n=1,2,..
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