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?
Make solar eclipses exceedingly rare, but still have new moons
Is it convenient to ask the journal's editor for two additional days to complete a review?
Can we say or write : "No, it'sn't"?
How is this set of matrices closed under multiplication?
Why do remote US companies require working in the US?
Is there always a complete, orthogonal set of unitary matrices?
Why, when going from special to general relativity, do we just replace partial derivatives with covariant derivatives?
Flying from Cape Town to England and return to another province
What connection does MS Office have to Netscape Navigator?
Grabbing quick drinks
How to edit “Name” property in GCI output?
If Nick Fury and Coulson already knew about aliens (Kree and Skrull) why did they wait until Thor's appearance to start making weapons?
Find non-case sensitive string in a mixed list of elements?
Won the lottery - how do I keep the money?
How many extra stops do monopods offer for tele photographs?
How to check if all elements of 1 list are in the *same quantity* and in any order, in the list2?
Why isn't acceleration always zero whenever velocity is zero, such as the moment a ball bounces off a wall?
What happened in Rome, when the western empire "fell"?
A small doubt about the dominated convergence theorem
How did people program for Consoles with multiple CPUs?
How do I align (1) and (2)?
How to write a definition with variants?
When you upcast Blindness/Deafness, do all targets suffer the same effect?
Is it possible to replace duplicates of a character with one character using tr
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 2 hours ago
Maarten BaertMaarten Baert
132
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 2 hours ago
Maarten BaertMaarten Baert
132
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 2 hours ago
Maarten BaertMaarten Baert
132
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 2 hours ago
Maarten BaertMaarten Baert
132
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 2 hours ago
Maarten BaertMaarten Baert
132
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 2 hours ago
Maarten BaertMaarten Baert
132
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 2 hours ago
Maarten BaertMaarten Baert
132
asked 2 hours ago
Maarten BaertMaarten Baert
132
132
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 2 hours ago
coreyman317coreyman317
1,069420
$endgroup$
$begingroup$
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
1 hour 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
1 hour 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 1 hour 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 2 hours ago
coreyman317coreyman317
1,069420
$endgroup$
$begingroup$
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
1 hour 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
1 hour 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 2 hours ago
coreyman317coreyman317
1,069420
$endgroup$
$begingroup$
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
1 hour 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
1 hour 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 2 hours ago
coreyman317coreyman317
1,069420
$endgroup$
$$y=frac1-2e^-xcos(x)+e^-2x1+2e^-xsin(x)-e^-2xcdotfrace^xe^x=frace^x-2cos(x)+e^-xe^x+2sin(x)-e^-xcdotfracfrac12frac12$$ $$=fracfrace^x+e^-x2-cos(x)frace^x-e^-x2+sin(x)=fraccosh(x)-cos(x)sinh(x)+sin(x)$$
answered 2 hours ago
coreyman317coreyman317
1,069420
answered 2 hours ago
coreyman317coreyman317
1,069420
answered 2 hours ago
coreyman317coreyman317
1,069420
answered 2 hours ago
coreyman317coreyman317
1,069420
1,069420
$begingroup$
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
1 hour 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
1 hour 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
1 hour 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
1 hour 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
1 hour 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
1 hour 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
1 hour 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
1 hour 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 1 hour 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 1 hour 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 1 hour 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 1 hour ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 1 hour ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 1 hour ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 1 hour 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
