Can this equation be simplified further? The Next CEO of Stack OverflowHow can this trig equation be simplified?Can this be simplified?How can this equation be simplified this way? Transmission line: ZinTaking the integral of a strange function.Evaluating Complex ExpressionsHow should trigonometric expressions be simplified?How can we show $cos^6x+sin^6x=1-3sin^2x cos^2x$?Roots of complex quadratic polynomialCan $Asin^2t + Bsin tcos t + Csin t + Dcos t + E = 0$ be solved algebraically?How was this equation simplified?
The past simple of "gaslight" – "gaslighted" or "gaslit"?
WOW air has ceased operation, can I get my tickets refunded?
Is it ever safe to open a suspicious HTML file (e.g. email attachment)?
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
How to get from Geneva Airport to Metabief, Doubs, France by public transport?
What connection does MS Office have to Netscape Navigator?
Reference request: Grassmannian and Plucker coordinates in type B, C, D
Prepend last line of stdin to entire stdin
Is wanting to ask what to write an indication that you need to change your story?
Is it professional to write unrelated content in an almost-empty email?
Grabbing quick drinks
Bartok - Syncopation (1): Meaning of notes in between Grand Staff
Would a grinding machine be a simple and workable propulsion system for an interplanetary spacecraft?
Running a General Election and the European Elections together
If Nick Fury and Coulson already knew about aliens (Kree and Skrull) why did they wait until Thor's appearance to start making weapons?
Why did CATV standarize in 75 ohms and everyone else in 50?
A Man With a Stainless Steel Endoskeleton (like The Terminator) Fighting Cloaked Aliens Only He Can See
Is there always a complete, orthogonal set of unitary matrices?
Why don't programming languages automatically manage the synchronous/asynchronous problem?
Is it okay to majorly distort historical facts while writing a fiction story?
Example of a Mathematician/Physicist whose Other Publications during their PhD eclipsed their PhD Thesis
How many extra stops do monopods offer for tele photographs?
Why is the US ranked as #45 in Press Freedom ratings, despite its extremely permissive free speech laws?
Is a distribution that is normal, but highly skewed considered Gaussian?
Can this equation be simplified further?
The Next CEO of Stack OverflowHow can this trig equation be simplified?Can this be simplified?How can this equation be simplified this way? Transmission line: ZinTaking the integral of a strange function.Evaluating Complex ExpressionsHow should trigonometric expressions be simplified?How can we show $cos^6x+sin^6x=1-3sin^2x cos^2x$?Roots of complex quadratic polynomialCan $Asin^2t + Bsin tcos t + Csin t + Dcos t + E = 0$ be solved algebraically?How was this equation simplified?
$begingroup$
I'm trying to simplify the following equation:
$y = dfrac1-2exp(-x)cos(x)+exp(-2x)1+2exp(-x)sin(x)-exp(-2x)$
I suspect that a simpler form using complex exponents exists, but I can't find it.
For context, this equation describes the effective conductivity due to the skin effect of a flat conductor as a function of its thickness. I just removed some scale factors for simplicity. The underlying differential equation gives rise to expressions of the form $exp(pm(1+i)x)$, which is where the $sin(x)$ and $cos(x)$ came from.
trigonometry complex-numbers
asked 1 hour ago
Maarten BaertMaarten Baert
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I'm trying to simplify the following equation:
$y = dfrac1-2exp(-x)cos(x)+exp(-2x)1+2exp(-x)sin(x)-exp(-2x)$
I suspect that a simpler form using complex exponents exists, but I can't find it.
For context, this equation describes the effective conductivity due to the skin effect of a flat conductor as a function of its thickness. I just removed some scale factors for simplicity. The underlying differential equation gives rise to expressions of the form $exp(pm(1+i)x)$, which is where the $sin(x)$ and $cos(x)$ came from.
trigonometry complex-numbers
asked 1 hour ago
Maarten BaertMaarten Baert
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I'm trying to simplify the following equation:
$y = dfrac1-2exp(-x)cos(x)+exp(-2x)1+2exp(-x)sin(x)-exp(-2x)$
I suspect that a simpler form using complex exponents exists, but I can't find it.
For context, this equation describes the effective conductivity due to the skin effect of a flat conductor as a function of its thickness. I just removed some scale factors for simplicity. The underlying differential equation gives rise to expressions of the form $exp(pm(1+i)x)$, which is where the $sin(x)$ and $cos(x)$ came from.
trigonometry complex-numbers
asked 1 hour ago
Maarten BaertMaarten Baert
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I'm trying to simplify the following equation:
$y = dfrac1-2exp(-x)cos(x)+exp(-2x)1+2exp(-x)sin(x)-exp(-2x)$
I suspect that a simpler form using complex exponents exists, but I can't find it.
For context, this equation describes the effective conductivity due to the skin effect of a flat conductor as a function of its thickness. I just removed some scale factors for simplicity. The underlying differential equation gives rise to expressions of the form $exp(pm(1+i)x)$, which is where the $sin(x)$ and $cos(x)$ came from.
trigonometry complex-numbers
trigonometry complex-numbers
asked 1 hour ago
Maarten BaertMaarten Baert
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Maarten BaertMaarten Baert
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Maarten BaertMaarten Baert
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Maarten BaertMaarten Baert
82
asked 1 hour ago
Maarten BaertMaarten Baert
82
82
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Maarten Baert is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$$y=frac1-2e^-xcos(x)+e^-2x1+2e^-xsin(x)-e^-2xcdotfrace^xe^x=frace^x-2cos(x)+e^-xe^x+2sin(x)-e^-xcdotfracfrac12frac12$$ $$=fracfrace^x+e^-x2-cos(x)frace^x-e^-x2+sin(x)=fraccosh(x)-cos(x)sinh(x)+sin(x)$$
answered 1 hour ago
coreyman317coreyman317
1,059420
$endgroup$
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
1
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
add a comment |
$begingroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$fraccosh(x)-cos(x)sinh(x)+sin(x) sim coth(x)$$ for an error $ < 10^-10$
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
$endgroup$
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
);
);
Maarten Baert is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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%2f3168988%2fcan-this-equation-be-simplified-further%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$y=frac1-2e^-xcos(x)+e^-2x1+2e^-xsin(x)-e^-2xcdotfrace^xe^x=frace^x-2cos(x)+e^-xe^x+2sin(x)-e^-xcdotfracfrac12frac12$$ $$=fracfrace^x+e^-x2-cos(x)frace^x-e^-x2+sin(x)=fraccosh(x)-cos(x)sinh(x)+sin(x)$$
answered 1 hour ago
coreyman317coreyman317
1,059420
$endgroup$
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
1
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
add a comment |
$begingroup$
$$y=frac1-2e^-xcos(x)+e^-2x1+2e^-xsin(x)-e^-2xcdotfrace^xe^x=frace^x-2cos(x)+e^-xe^x+2sin(x)-e^-xcdotfracfrac12frac12$$ $$=fracfrace^x+e^-x2-cos(x)frace^x-e^-x2+sin(x)=fraccosh(x)-cos(x)sinh(x)+sin(x)$$
answered 1 hour ago
coreyman317coreyman317
1,059420
$endgroup$
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
1
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
add a comment |
$begingroup$
$$y=frac1-2e^-xcos(x)+e^-2x1+2e^-xsin(x)-e^-2xcdotfrace^xe^x=frace^x-2cos(x)+e^-xe^x+2sin(x)-e^-xcdotfracfrac12frac12$$ $$=fracfrace^x+e^-x2-cos(x)frace^x-e^-x2+sin(x)=fraccosh(x)-cos(x)sinh(x)+sin(x)$$
answered 1 hour ago
coreyman317coreyman317
1,059420
$endgroup$
$$y=frac1-2e^-xcos(x)+e^-2x1+2e^-xsin(x)-e^-2xcdotfrace^xe^x=frace^x-2cos(x)+e^-xe^x+2sin(x)-e^-xcdotfracfrac12frac12$$ $$=fracfrace^x+e^-x2-cos(x)frace^x-e^-x2+sin(x)=fraccosh(x)-cos(x)sinh(x)+sin(x)$$
answered 1 hour ago
coreyman317coreyman317
1,059420
answered 1 hour ago
coreyman317coreyman317
1,059420
answered 1 hour ago
coreyman317coreyman317
1,059420
answered 1 hour ago
coreyman317coreyman317
1,059420
1,059420
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
1
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
add a comment |
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
1
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
$begingroup$
Nice! Would it be possible to rewrite this using $tan$ or $tanh$? Unfortunately $sinh(x)$ and $cosh(x)$ cause numerical issues (overflow) for large values of $x$.
$endgroup$
– Maarten Baert
54 mins ago
1
1
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
$begingroup$
If you want you can divide the top and bottom by $cosh(x)$ to get a $tanh(x)$ but this makes both the numerator and denominator more complicated.
$endgroup$
– coreyman317
50 mins ago
add a comment |
$begingroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$fraccosh(x)-cos(x)sinh(x)+sin(x) sim coth(x)$$ for an error $ < 10^-10$
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
$endgroup$
add a comment |
$begingroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$fraccosh(x)-cos(x)sinh(x)+sin(x) sim coth(x)$$ for an error $ < 10^-10$
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
$endgroup$
add a comment |
$begingroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$fraccosh(x)-cos(x)sinh(x)+sin(x) sim coth(x)$$ for an error $ < 10^-10$
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
$endgroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$fraccosh(x)-cos(x)sinh(x)+sin(x) sim coth(x)$$ for an error $ < 10^-10$
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 34 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
125k1158135
add a comment |
add a comment |
Maarten Baert is a new contributor. Be nice, and check out our Code of Conduct.
Maarten Baert is a new contributor. Be nice, and check out our Code of Conduct.
Maarten Baert is a new contributor. Be nice, and check out our Code of Conduct.
Maarten Baert is a new contributor. Be nice, and check out our Code of Conduct.
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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%2f3168988%2fcan-this-equation-be-simplified-further%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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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
