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Can this equation be simplified further?
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$begingroup$
I'm trying to simplify the following equation:
$y = dfrac{1-2exp(-x)cos(x)+exp(-2x)}{1+2exp(-x)sin(x)-exp(-2x)}$
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:
$y = dfrac{1-2exp(-x)cos(x)+exp(-2x)}{1+2exp(-x)sin(x)-exp(-2x)}$
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:
$y = dfrac{1-2exp(-x)cos(x)+exp(-2x)}{1+2exp(-x)sin(x)-exp(-2x)}$
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:
$y = dfrac{1-2exp(-x)cos(x)+exp(-2x)}{1+2exp(-x)sin(x)-exp(-2x)}$
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
$begingroup$
$$y=frac{1-2e^{-x}cos(x)+e^{-2x}}{1+2e^{-x}sin(x)-e^{-2x}}cdotfrac{e^x}{e^x}=frac{e^x-2cos(x)+e^{-x}}{e^x+2sin(x)-e^{-x}}cdotfrac{frac{1}{2}}{frac{1}{2}}$$ $$=frac{frac{e^x+e^{-x}}{2}-cos(x)}{frac{e^x-e^{-x}}{2}+sin(x)}=frac{cosh(x)-cos(x)}{sinh(x)+sin(x)}$$
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
58 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
54 mins ago
add a comment |
$begingroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$frac{cosh(x)-cos(x)}{sinh(x)+sin(x)} sim coth(x)$$ for an error $ < 10^{-10}$
answered 38 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$y=frac{1-2e^{-x}cos(x)+e^{-2x}}{1+2e^{-x}sin(x)-e^{-2x}}cdotfrac{e^x}{e^x}=frac{e^x-2cos(x)+e^{-x}}{e^x+2sin(x)-e^{-x}}cdotfrac{frac{1}{2}}{frac{1}{2}}$$ $$=frac{frac{e^x+e^{-x}}{2}-cos(x)}{frac{e^x-e^{-x}}{2}+sin(x)}=frac{cosh(x)-cos(x)}{sinh(x)+sin(x)}$$
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
58 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
54 mins ago
add a comment |
$begingroup$
$$y=frac{1-2e^{-x}cos(x)+e^{-2x}}{1+2e^{-x}sin(x)-e^{-2x}}cdotfrac{e^x}{e^x}=frac{e^x-2cos(x)+e^{-x}}{e^x+2sin(x)-e^{-x}}cdotfrac{frac{1}{2}}{frac{1}{2}}$$ $$=frac{frac{e^x+e^{-x}}{2}-cos(x)}{frac{e^x-e^{-x}}{2}+sin(x)}=frac{cosh(x)-cos(x)}{sinh(x)+sin(x)}$$
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
58 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
54 mins ago
add a comment |
$begingroup$
$$y=frac{1-2e^{-x}cos(x)+e^{-2x}}{1+2e^{-x}sin(x)-e^{-2x}}cdotfrac{e^x}{e^x}=frac{e^x-2cos(x)+e^{-x}}{e^x+2sin(x)-e^{-x}}cdotfrac{frac{1}{2}}{frac{1}{2}}$$ $$=frac{frac{e^x+e^{-x}}{2}-cos(x)}{frac{e^x-e^{-x}}{2}+sin(x)}=frac{cosh(x)-cos(x)}{sinh(x)+sin(x)}$$
answered 1 hour ago
coreyman317coreyman317
1,059420
$endgroup$
$$y=frac{1-2e^{-x}cos(x)+e^{-2x}}{1+2e^{-x}sin(x)-e^{-2x}}cdotfrac{e^x}{e^x}=frac{e^x-2cos(x)+e^{-x}}{e^x+2sin(x)-e^{-x}}cdotfrac{frac{1}{2}}{frac{1}{2}}$$ $$=frac{frac{e^x+e^{-x}}{2}-cos(x)}{frac{e^x-e^{-x}}{2}+sin(x)}=frac{cosh(x)-cos(x)}{sinh(x)+sin(x)}$$
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
58 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
54 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
58 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
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
58 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
58 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
54 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
54 mins ago
add a comment |
$begingroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$frac{cosh(x)-cos(x)}{sinh(x)+sin(x)} sim coth(x)$$ for an error $ < 10^{-10}$
answered 38 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 $x >24$
$$frac{cosh(x)-cos(x)}{sinh(x)+sin(x)} sim coth(x)$$ for an error $ < 10^{-10}$
answered 38 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 $x >24$
$$frac{cosh(x)-cos(x)}{sinh(x)+sin(x)} sim coth(x)$$ for an error $ < 10^{-10}$
answered 38 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
$endgroup$
After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$frac{cosh(x)-cos(x)}{sinh(x)+sin(x)} sim coth(x)$$ for an error $ < 10^{-10}$
answered 38 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 38 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 38 mins ago
Claude LeiboviciClaude Leibovici
125k1158135
answered 38 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.
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