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I'm trying to simplify the following equation:

$y = \dfrac{1-2\exp(-x)\cos(x)+\exp(-2x)}{1+2\exp(-x)\sin(x)-\exp(-2x)}$

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.

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$$y=\frac{1-2e^{-x}\cos(x)+e^{-2x}}{1+2e^{-x}\sin(x)-e^{-2x}}\cdot\frac{e^x}{e^x}=\frac{e^x-2\cos(x)+e^{-x}}{e^x+2\sin(x)-e^{-x}}\cdot\frac{\frac{1}{2}}{\frac{1}{2}}$$ $$=\frac{\frac{e^x+e^{-x}}{2}-\cos(x)}{\frac{e^x-e^{-x}}{2}+\sin(x)}=\frac{\cosh(x)-\cos(x)}{\sinh(x)+\sin(x)}$$

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  • $\begingroup$ 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 Mar 31 '19 at 3:21
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    $\begingroup$ 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 Mar 31 '19 at 3:25
  • $\begingroup$ @coreyman317 you from san diego? $\endgroup$ – user686544 Aug 16 '19 at 2:02
  • $\begingroup$ Unfortunately not; I'm from the Midwest. @Eastcoastdancer $\endgroup$ – coreyman317 Aug 16 '19 at 2:02
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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}$.

Moreover, for large $x$, $\coth(x)\sim 1+2e^{-2x}$.

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