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
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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,..](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4274ade8-8ad2-4d14-88b8-423376a3a942%2F8fca2b40-e048-44d7-8bc1-2d4519380ec4%2Fr47ewyv_processed.jpeg&w=3840&q=75)
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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