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Symmetries in Fourier series



The first two are trivial standard results. However i'm unsure how to approach the remainder.

For the coefficients to be zero we need f(x)cos(n*pi*x/l) or f(x)sin(n*pi*x/l) to be odd within the period |x|< l .

for instance in c) without knowing the parity of f(x)=f(pi-x) I'm not sure where to go.
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TeeEm
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Original post by Goods


The first two are trivial standard results. However i'm unsure how to approach the remainder.

For the coefficients to be zero we need f(x)cos(n*pi*x/l) or f(x)sin(n*pi*x/l) to be odd within the period |x|< l .

for instance in c) without knowing the parity of f(x)=f(pi-x) I'm not sure where to go.


for practical purposes these symmetries are almost useless... I personally never learned them.
if you want to derive them think of the compound angle for sine and cosine
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Goods
OP
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Original post by TeeEm
for practical purposes these symmetries are almost useless... I personally never learned them.
if you want to derive them think of the compound angle for sine and cosine


Doing that would give me whether or not the trig term was odd or even for given values of n. But if i don't know where f(x) is odd or even I still can't say if an or bn would be zero?
Note that if you know ∑0∞ancos⁡nx+bnsin⁡nx=∑0∞cncos⁡nx+dnsin⁡nx\sum_0^\infty a_n \cos nx + b_n \sin nx = \sum_0^\infty c_n \cos nx+ d_n \sin nx, then we must have

an=cn, bn=dn∀na_n = c_n, \, b_n = d_n \forall n. This should be enough for you to finish the question.
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Goods
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Original post by DFranklin
Note that if you know ∑0∞ancos⁡nx+bnsin⁡nx=∑0∞cncos⁡nx+dnsin⁡nx\sum_0^\infty a_n \cos nx + b_n \sin nx = \sum_0^\infty c_n \cos nx+ d_n \sin nx, then we must have

an=cn, bn=dn∀na_n = c_n, \, b_n = d_n \forall n. This should be enough for you to finish the question.


*What i suggested was wrong will try again*


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(edited 9 years ago)
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Goods
OP
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Original post by DFranklin
Note that if you know ∑0∞ancos⁡nx+bnsin⁡nx=∑0∞cncos⁡nx+dnsin⁡nx\sum_0^\infty a_n \cos nx + b_n \sin nx = \sum_0^\infty c_n \cos nx+ d_n \sin nx, then we must have

an=cn, bn=dn∀na_n = c_n, \, b_n = d_n \forall n. This should be enough for you to finish the question.


I feel I must be missing something really obvious.. could you hint where to go?
Original post by Goods
I feel I must be missing something really obvious.. could you hint where to go?


I think DFranklin is meaning if you write out the series for f(x) and the series for [in part c, for example] f(pi-x) and set them equal, the coefficients must be the same. But you can simplify f(pi-x) [like] so this should help you see what the coefficients are.

So you basically write out the full series for [eg] f(x)=f(pi-x). f(x) is already written out for you in the question, and just change the x to pi-x to write out the series for f(pi-x).
(edited 9 years ago)
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Goods
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Original post by DFranklin


Original post by rayquaza17


Original post by TeeEm


Thanks so much for the help I can see how to to work them out them now!

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