Prove an equality of Fourier series.
$begingroup$
Consider the function:
$f(x)= left{ begin{array}{lcc}
frac{3x}{2} & if & 0leq x leq frac{pi}{3} \
\ frac{pi}{2} & si & frac{pi}3 < x < frac{2pi}3 \
\ frac{3(pi - x)}{2} & si & frac{2pi}{3}leq x leq pi
end{array}
right.$
Prove
$$f(x)=sum_{i=1}^inftyfrac{sin(2n-1)pi}{3(2n-1)^2}times sin(2n-1)x,,,, text{for 0$leq$x$leqpi$ }$$
My attempt:
We need find the representation of fourier series in $sin$ of $f$.
As $f$ is continuous then
$$f(x)=sum_{i=1}^nb_ntimes sin(nx)$ with $xin[0,pi]$$
where,
$$b_n=frac{2}{pi}int_{0}^{pi} f(x)times sin(nx)dx$$
Using Maple I have:
$$b_n=frac{2}{pi}[frac{3}{2},{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}
]$$
then,
$$b_n=frac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]$$
this implies:
$$f(x)=sum_{i=1}^nfrac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]sin(nx)$$ for $0leq xleq pi$
here, I'm stuck. can someone help me?
fourier-series
edited Dec 29 '18 at 16:20
Bernard
119k639112
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
$endgroup$
add a comment |
$begingroup$
Consider the function:
$f(x)= left{ begin{array}{lcc}
frac{3x}{2} & if & 0leq x leq frac{pi}{3} \
\ frac{pi}{2} & si & frac{pi}3 < x < frac{2pi}3 \
\ frac{3(pi - x)}{2} & si & frac{2pi}{3}leq x leq pi
end{array}
right.$
Prove
$$f(x)=sum_{i=1}^inftyfrac{sin(2n-1)pi}{3(2n-1)^2}times sin(2n-1)x,,,, text{for 0$leq$x$leqpi$ }$$
My attempt:
We need find the representation of fourier series in $sin$ of $f$.
As $f$ is continuous then
$$f(x)=sum_{i=1}^nb_ntimes sin(nx)$ with $xin[0,pi]$$
where,
$$b_n=frac{2}{pi}int_{0}^{pi} f(x)times sin(nx)dx$$
Using Maple I have:
$$b_n=frac{2}{pi}[frac{3}{2},{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}
]$$
then,
$$b_n=frac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]$$
this implies:
$$f(x)=sum_{i=1}^nfrac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]sin(nx)$$ for $0leq xleq pi$
here, I'm stuck. can someone help me?
fourier-series
edited Dec 29 '18 at 16:20
Bernard
119k639112
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
$endgroup$
add a comment |
$begingroup$
Consider the function:
$f(x)= left{ begin{array}{lcc}
frac{3x}{2} & if & 0leq x leq frac{pi}{3} \
\ frac{pi}{2} & si & frac{pi}3 < x < frac{2pi}3 \
\ frac{3(pi - x)}{2} & si & frac{2pi}{3}leq x leq pi
end{array}
right.$
Prove
$$f(x)=sum_{i=1}^inftyfrac{sin(2n-1)pi}{3(2n-1)^2}times sin(2n-1)x,,,, text{for 0$leq$x$leqpi$ }$$
My attempt:
We need find the representation of fourier series in $sin$ of $f$.
As $f$ is continuous then
$$f(x)=sum_{i=1}^nb_ntimes sin(nx)$ with $xin[0,pi]$$
where,
$$b_n=frac{2}{pi}int_{0}^{pi} f(x)times sin(nx)dx$$
Using Maple I have:
$$b_n=frac{2}{pi}[frac{3}{2},{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}
]$$
then,
$$b_n=frac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]$$
this implies:
$$f(x)=sum_{i=1}^nfrac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]sin(nx)$$ for $0leq xleq pi$
here, I'm stuck. can someone help me?
fourier-series
edited Dec 29 '18 at 16:20
Bernard
119k639112
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
$endgroup$
Consider the function:
$f(x)= left{ begin{array}{lcc}
frac{3x}{2} & if & 0leq x leq frac{pi}{3} \
\ frac{pi}{2} & si & frac{pi}3 < x < frac{2pi}3 \
\ frac{3(pi - x)}{2} & si & frac{2pi}{3}leq x leq pi
end{array}
right.$
Prove
$$f(x)=sum_{i=1}^inftyfrac{sin(2n-1)pi}{3(2n-1)^2}times sin(2n-1)x,,,, text{for 0$leq$x$leqpi$ }$$
My attempt:
We need find the representation of fourier series in $sin$ of $f$.
As $f$ is continuous then
$$f(x)=sum_{i=1}^nb_ntimes sin(nx)$ with $xin[0,pi]$$
where,
$$b_n=frac{2}{pi}int_{0}^{pi} f(x)times sin(nx)dx$$
Using Maple I have:
$$b_n=frac{2}{pi}[frac{3}{2},{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}
]$$
then,
$$b_n=frac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]$$
this implies:
$$f(x)=sum_{i=1}^nfrac{1}{pi}[3,{frac {sin left( frac{pi n}{3} right) -sin left( pi ,n
right) +sin left( frac{2 pi n}{3} right) }{{n}^{2}}}]sin(nx)$$ for $0leq xleq pi$
here, I'm stuck. can someone help me?
fourier-series
fourier-series
edited Dec 29 '18 at 16:20
Bernard
119k639112
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
edited Dec 29 '18 at 16:20
Bernard
119k639112
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
edited Dec 29 '18 at 16:20
Bernard
119k639112
edited Dec 29 '18 at 16:20
Bernard
119k639112
edited Dec 29 '18 at 16:20
Bernard
119k639112
119k639112
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
asked Dec 29 '18 at 16:16
Bvss12Bvss12
1,778617
1,778617
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055995%2fprove-an-equality-of-fourier-series%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055995%2fprove-an-equality-of-fourier-series%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
