Confusion regarding Fourier half series for sine and cosine The 2019 Stack Overflow Developer Survey Results Are InRelation between “harmonic form” and fourier series?Fourier cosine series and sum helpWhat is a fast method for evaluate this trigonometric series?Fourier series with half rangeFourier Cosine Series questionFourier cosine series giving nonsense answerFourier Series. Relevance of the period of sine and cosineUndefined term in Fourier seriesFourier Series Expansion for Half-Wave Sine ProblemComplex sum of sine and cosine function
Why was M87 targeted for the Event Horizon Telescope instead of Sagittarius A*?
How do PCB vias affect signal quality?
What do I do when my TA workload is more than expected?
What to do when moving next to a bird sanctuary with a loosely-domesticated cat?
How did passengers keep warm on sail ships?
Output the Arecibo Message
Will it cause any balance problems to have PCs level up and gain the benefits of a long rest mid-fight?
A word that means fill it to the required quantity
Correct punctuation for showing a character's confusion
Is it ethical to upload a automatically generated paper to a non peer-reviewed site as part of a larger research?
Why can't devices on different VLANs, but on the same subnet, communicate?
Why is the maximum length of OpenWrt’s root password 8 characters?
Cooking pasta in a water boiler
Is bread bad for ducks?
Is it ok to offer lower paid work as a trial period before negotiating for a full-time job?
What is the motivation for a law requiring 2 parties to consent for recording a conversation
For what reasons would an animal species NOT cross a *horizontal* land bridge?
Did any laptop computers have a built-in 5 1/4 inch floppy drive?
The phrase "to the numbers born"?
Can an undergraduate be advised by a professor who is very far away?
Why don't hard Brexiteers insist on a hard border to prevent illegal immigration after Brexit?
Are spiders unable to hurt humans, especially very small spiders?
How to translate "being like"?
Short story: man watches girlfriend's spaceship entering a 'black hole' (?) forever
Confusion regarding Fourier half series for sine and cosine
The 2019 Stack Overflow Developer Survey Results Are InRelation between “harmonic form” and fourier series?Fourier cosine series and sum helpWhat is a fast method for evaluate this trigonometric series?Fourier series with half rangeFourier Cosine Series questionFourier cosine series giving nonsense answerFourier Series. Relevance of the period of sine and cosineUndefined term in Fourier seriesFourier Series Expansion for Half-Wave Sine ProblemComplex sum of sine and cosine function
$begingroup$
I have been struggling with a problem for a long time. Solving a second order partial differential equation using Fourier half series in sine with the help of Mathematica gives me
$$
phi_mine(x,y)=-sum_k=1,3,5,...^inftyfrac8 a^2 G_zy theta left(textsechleft(fracpi b k2 a fracsqrtG_zxsqrtG_zy right) cosh left(fracpi k ya fracsqrtG_zxsqrtG_zyright)-1right)pi ^3 k^3sin left(fracpi k xaright)
$$
where $G_zy$, $G_zx$, $theta$, $a$, and $b$ are constants.
This is almost exactly what is written in the solution that I have, which is
$$
phi_sol(x,y)=frac8pi^3 G_zy a^2 sum_k=1,3,5,...^inftyfrac(-1)^(k-1)/2k^3left( 1-fraccosh left(fracpi k mu ayright)cosh left(fracb pi k mu2 aright) right)cos left(fracpi kaxright)
$$
where $mu=sqrtfracG_zxG_zy$.
$phi_sol(x,y)$ is missing $theta$ but I suspect that it's a typo. I've been trying to figure out the difference between my answer and the solution and all I can find is that somehow
$$
sin left(fracpi k xaright)=cosleft(fracpi k xaright) (-1)^(k-1)/2
$$
for $k=1,3,5,...$ I've never seen this and when I plot them they give different curves so they don't seem to be equivalent.
As can be seen, $a$ determines the amplitude of the periodic functions so that $
sin left(fracpi k xaright)$ is always $0$ at -1 and 1. $cosleft(fracpi k xaright) (-1)^(k-1)/2$ on the other hand flips from $1$ to $-1$ at $pm a$.
The next step in the process involves working out the constant $beta$,
$$
beta=frac2 int_0^b left(int_0^a phi (x,y) , dxright) , dyG_zx a b^3
$$
I get that $phi_sol(x,y)$ converges to a value while $phi_mine(x,y)$ goes to infinity. Therefore I'm doing something wrong but I'm not sure what. Is it possible to change $sin left(fracpi k xaright)$ into $cosleft(fracpi k xaright) (-1)^(k-1)/2$?
fourier-series trigonometric-series
asked Apr 7 at 10:52
enea19enea19
374
$endgroup$
add a comment |
$begingroup$
I have been struggling with a problem for a long time. Solving a second order partial differential equation using Fourier half series in sine with the help of Mathematica gives me
$$
phi_mine(x,y)=-sum_k=1,3,5,...^inftyfrac8 a^2 G_zy theta left(textsechleft(fracpi b k2 a fracsqrtG_zxsqrtG_zy right) cosh left(fracpi k ya fracsqrtG_zxsqrtG_zyright)-1right)pi ^3 k^3sin left(fracpi k xaright)
$$
where $G_zy$, $G_zx$, $theta$, $a$, and $b$ are constants.
This is almost exactly what is written in the solution that I have, which is
$$
phi_sol(x,y)=frac8pi^3 G_zy a^2 sum_k=1,3,5,...^inftyfrac(-1)^(k-1)/2k^3left( 1-fraccosh left(fracpi k mu ayright)cosh left(fracb pi k mu2 aright) right)cos left(fracpi kaxright)
$$
where $mu=sqrtfracG_zxG_zy$.
$phi_sol(x,y)$ is missing $theta$ but I suspect that it's a typo. I've been trying to figure out the difference between my answer and the solution and all I can find is that somehow
$$
sin left(fracpi k xaright)=cosleft(fracpi k xaright) (-1)^(k-1)/2
$$
for $k=1,3,5,...$ I've never seen this and when I plot them they give different curves so they don't seem to be equivalent.
As can be seen, $a$ determines the amplitude of the periodic functions so that $
sin left(fracpi k xaright)$ is always $0$ at -1 and 1. $cosleft(fracpi k xaright) (-1)^(k-1)/2$ on the other hand flips from $1$ to $-1$ at $pm a$.
The next step in the process involves working out the constant $beta$,
$$
beta=frac2 int_0^b left(int_0^a phi (x,y) , dxright) , dyG_zx a b^3
$$
I get that $phi_sol(x,y)$ converges to a value while $phi_mine(x,y)$ goes to infinity. Therefore I'm doing something wrong but I'm not sure what. Is it possible to change $sin left(fracpi k xaright)$ into $cosleft(fracpi k xaright) (-1)^(k-1)/2$?
fourier-series trigonometric-series
asked Apr 7 at 10:52
enea19enea19
374
$endgroup$
$begingroup$
No, that's wrong. I corrected it, thanks for pointing it out.
$endgroup$
– enea19
Apr 8 at 1:07
add a comment |
$begingroup$
I have been struggling with a problem for a long time. Solving a second order partial differential equation using Fourier half series in sine with the help of Mathematica gives me
$$
phi_mine(x,y)=-sum_k=1,3,5,...^inftyfrac8 a^2 G_zy theta left(textsechleft(fracpi b k2 a fracsqrtG_zxsqrtG_zy right) cosh left(fracpi k ya fracsqrtG_zxsqrtG_zyright)-1right)pi ^3 k^3sin left(fracpi k xaright)
$$
where $G_zy$, $G_zx$, $theta$, $a$, and $b$ are constants.
This is almost exactly what is written in the solution that I have, which is
$$
phi_sol(x,y)=frac8pi^3 G_zy a^2 sum_k=1,3,5,...^inftyfrac(-1)^(k-1)/2k^3left( 1-fraccosh left(fracpi k mu ayright)cosh left(fracb pi k mu2 aright) right)cos left(fracpi kaxright)
$$
where $mu=sqrtfracG_zxG_zy$.
$phi_sol(x,y)$ is missing $theta$ but I suspect that it's a typo. I've been trying to figure out the difference between my answer and the solution and all I can find is that somehow
$$
sin left(fracpi k xaright)=cosleft(fracpi k xaright) (-1)^(k-1)/2
$$
for $k=1,3,5,...$ I've never seen this and when I plot them they give different curves so they don't seem to be equivalent.
As can be seen, $a$ determines the amplitude of the periodic functions so that $
sin left(fracpi k xaright)$ is always $0$ at -1 and 1. $cosleft(fracpi k xaright) (-1)^(k-1)/2$ on the other hand flips from $1$ to $-1$ at $pm a$.
The next step in the process involves working out the constant $beta$,
$$
beta=frac2 int_0^b left(int_0^a phi (x,y) , dxright) , dyG_zx a b^3
$$
I get that $phi_sol(x,y)$ converges to a value while $phi_mine(x,y)$ goes to infinity. Therefore I'm doing something wrong but I'm not sure what. Is it possible to change $sin left(fracpi k xaright)$ into $cosleft(fracpi k xaright) (-1)^(k-1)/2$?
fourier-series trigonometric-series
asked Apr 7 at 10:52
enea19enea19
374
$endgroup$
I have been struggling with a problem for a long time. Solving a second order partial differential equation using Fourier half series in sine with the help of Mathematica gives me
$$
phi_mine(x,y)=-sum_k=1,3,5,...^inftyfrac8 a^2 G_zy theta left(textsechleft(fracpi b k2 a fracsqrtG_zxsqrtG_zy right) cosh left(fracpi k ya fracsqrtG_zxsqrtG_zyright)-1right)pi ^3 k^3sin left(fracpi k xaright)
$$
where $G_zy$, $G_zx$, $theta$, $a$, and $b$ are constants.
This is almost exactly what is written in the solution that I have, which is
$$
phi_sol(x,y)=frac8pi^3 G_zy a^2 sum_k=1,3,5,...^inftyfrac(-1)^(k-1)/2k^3left( 1-fraccosh left(fracpi k mu ayright)cosh left(fracb pi k mu2 aright) right)cos left(fracpi kaxright)
$$
where $mu=sqrtfracG_zxG_zy$.
$phi_sol(x,y)$ is missing $theta$ but I suspect that it's a typo. I've been trying to figure out the difference between my answer and the solution and all I can find is that somehow
$$
sin left(fracpi k xaright)=cosleft(fracpi k xaright) (-1)^(k-1)/2
$$
for $k=1,3,5,...$ I've never seen this and when I plot them they give different curves so they don't seem to be equivalent.
As can be seen, $a$ determines the amplitude of the periodic functions so that $
sin left(fracpi k xaright)$ is always $0$ at -1 and 1. $cosleft(fracpi k xaright) (-1)^(k-1)/2$ on the other hand flips from $1$ to $-1$ at $pm a$.
The next step in the process involves working out the constant $beta$,
$$
beta=frac2 int_0^b left(int_0^a phi (x,y) , dxright) , dyG_zx a b^3
$$
I get that $phi_sol(x,y)$ converges to a value while $phi_mine(x,y)$ goes to infinity. Therefore I'm doing something wrong but I'm not sure what. Is it possible to change $sin left(fracpi k xaright)$ into $cosleft(fracpi k xaright) (-1)^(k-1)/2$?
fourier-series trigonometric-series
fourier-series trigonometric-series
asked Apr 7 at 10:52
enea19enea19
374
asked Apr 7 at 10:52
enea19enea19
374
edited Apr 8 at 1:07
enea19
asked Apr 7 at 10:52
enea19enea19
374
asked Apr 7 at 10:52
enea19enea19
374
asked Apr 7 at 10:52
enea19enea19
374
374
$begingroup$
No, that's wrong. I corrected it, thanks for pointing it out.
$endgroup$
– enea19
Apr 8 at 1:07
add a comment |
$begingroup$
No, that's wrong. I corrected it, thanks for pointing it out.
$endgroup$
– enea19
Apr 8 at 1:07
$begingroup$
No, that's wrong. I corrected it, thanks for pointing it out.
$endgroup$
– enea19
Apr 8 at 1:07
$begingroup$
No, that's wrong. I corrected it, thanks for pointing it out.
$endgroup$
– enea19
Apr 8 at 1:07
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%2f3178067%2fconfusion-regarding-fourier-half-series-for-sine-and-cosine%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%2f3178067%2fconfusion-regarding-fourier-half-series-for-sine-and-cosine%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
$begingroup$
No, that's wrong. I corrected it, thanks for pointing it out.
$endgroup$
– enea19
Apr 8 at 1:07