Hyperbolic “Fourier” type infinite series for $log(sinh x)$ and $log(cosh x)$ The 2019 Stack Overflow Developer Survey Results Are InOn the zero of $f(x)=int_0^x ln(sinh(z))dz$Making use of Fourier series to evaluate an infinite sumCompute the fourier coefficients, and series for $log(sin(x))$Computing the series of log and sineProving $pi coth pi a= frac1a+ sum_n=1^inftyfrac2an^2+a^2$ using the Fourier series for $cosh ax$Fourier series, infinite seriesFourier series of: $[log(sin x)]^2$Fourier series for $cosh(x)$fourier series of $cosh(ax)$Find Fourier series of $cosh(ax)$Finding Fourier series by evaluating sum of infinite series
What are the motivations for publishing new editions of an existing textbook, beyond new discoveries in a field?
Delete all lines which don't have n characters before delimiter
Who coined the term "madman theory"?
How to support a colleague who finds meetings extremely tiring?
Should I use my personal e-mail address, or my workplace one, when registering to external websites for work purposes?
Are there any other methods to apply to solving simultaneous equations?
Can a rogue use sneak attack with weapons that have the thrown property even if they are not thrown?
Multiply Two Integer Polynomials
How can I autofill dates in Excel excluding Sunday?
Can we generate random numbers using irrational numbers like π and e?
Have you ever entered Singapore using a different passport or name?
Is a "Democratic" Oligarchy-Style System Possible?
What is the closest word meaning "respect for time / mindful"
For what reasons would an animal species NOT cross a *horizontal* land bridge?
Why isn't the circumferential light around the M87 black hole's event horizon symmetric?
One word riddle: Vowel in the middle
Can someone be penalized for an "unlawful" act if no penalty is specified?
Am I thawing this London Broil safely?
Shouldn't "much" here be used instead of "more"?
How to save as into a customized destination on macOS?
Aging parents with no investments
How to answer pointed "are you quitting" questioning when I don't want them to suspect
Write faster on AT24C32
Why do UK politicians seemingly ignore opinion polls on Brexit?
Hyperbolic “Fourier” type infinite series for $log(sinh x)$ and $log(cosh x)$
The 2019 Stack Overflow Developer Survey Results Are InOn the zero of $f(x)=int_0^x ln(sinh(z))dz$Making use of Fourier series to evaluate an infinite sumCompute the fourier coefficients, and series for $log(sin(x))$Computing the series of log and sineProving $pi coth pi a= frac1a+ sum_n=1^inftyfrac2an^2+a^2$ using the Fourier series for $cosh ax$Fourier series, infinite seriesFourier series of: $[log(sin x)]^2$Fourier series for $cosh(x)$fourier series of $cosh(ax)$Find Fourier series of $cosh(ax)$Finding Fourier series by evaluating sum of infinite series
$begingroup$
Hyperbolic Fourier type infinite series for $log(sinh x)$ and $log(cosh x)$ analogous to Fourier $cos$ series for $log(sin x)$ and $log(cos x)$
$$log(sinh(x))=-frac12 (i pi )-log (2)-sum _k=1^infty fraccosh (2 k x)k tag1$$
$$log(cosh(x))=-log (2)-sum _k=1^infty frac(-1)^k cosh (2 k x)k tag2$$
These analogous formula were found using Mathematica, but I am not sure about the proof. Do I convert $cosh(2kx)$ into its exponential form and proceed that way?
sequences-and-series fourier-series
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
$endgroup$
add a comment |
$begingroup$
Hyperbolic Fourier type infinite series for $log(sinh x)$ and $log(cosh x)$ analogous to Fourier $cos$ series for $log(sin x)$ and $log(cos x)$
$$log(sinh(x))=-frac12 (i pi )-log (2)-sum _k=1^infty fraccosh (2 k x)k tag1$$
$$log(cosh(x))=-log (2)-sum _k=1^infty frac(-1)^k cosh (2 k x)k tag2$$
These analogous formula were found using Mathematica, but I am not sure about the proof. Do I convert $cosh(2kx)$ into its exponential form and proceed that way?
sequences-and-series fourier-series
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
$endgroup$
add a comment |
$begingroup$
Hyperbolic Fourier type infinite series for $log(sinh x)$ and $log(cosh x)$ analogous to Fourier $cos$ series for $log(sin x)$ and $log(cos x)$
$$log(sinh(x))=-frac12 (i pi )-log (2)-sum _k=1^infty fraccosh (2 k x)k tag1$$
$$log(cosh(x))=-log (2)-sum _k=1^infty frac(-1)^k cosh (2 k x)k tag2$$
These analogous formula were found using Mathematica, but I am not sure about the proof. Do I convert $cosh(2kx)$ into its exponential form and proceed that way?
sequences-and-series fourier-series
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
$endgroup$
Hyperbolic Fourier type infinite series for $log(sinh x)$ and $log(cosh x)$ analogous to Fourier $cos$ series for $log(sin x)$ and $log(cos x)$
$$log(sinh(x))=-frac12 (i pi )-log (2)-sum _k=1^infty fraccosh (2 k x)k tag1$$
$$log(cosh(x))=-log (2)-sum _k=1^infty frac(-1)^k cosh (2 k x)k tag2$$
These analogous formula were found using Mathematica, but I am not sure about the proof. Do I convert $cosh(2kx)$ into its exponential form and proceed that way?
sequences-and-series fourier-series
sequences-and-series fourier-series
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
edited Jan 24 at 22:35
James Arathoon
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
asked Jan 24 at 22:21
James ArathoonJames Arathoon
1,608423
1,608423
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
$$sin(ix)=isinh(x)$$
and $$cos(ix)=cosh(x)$$
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
$endgroup$
add a comment |
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%2f3086436%2fhyperbolic-fourier-type-infinite-series-for-log-sinh-x-and-log-cosh-x%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
$$sin(ix)=isinh(x)$$
and $$cos(ix)=cosh(x)$$
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
$endgroup$
add a comment |
$begingroup$
Hint:
$$sin(ix)=isinh(x)$$
and $$cos(ix)=cosh(x)$$
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
$endgroup$
add a comment |
$begingroup$
Hint:
$$sin(ix)=isinh(x)$$
and $$cos(ix)=cosh(x)$$
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
$endgroup$
Hint:
$$sin(ix)=isinh(x)$$
and $$cos(ix)=cosh(x)$$
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
answered Mar 23 at 16:14
clathratusclathratus
5,1141439
5,1141439
add a comment |
add a comment |
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%2f3086436%2fhyperbolic-fourier-type-infinite-series-for-log-sinh-x-and-log-cosh-x%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
