Can this equation be simplified further?












1












$begingroup$


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







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    1












    $begingroup$


    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







    share|cite|improve this question







    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.







    $endgroup$















      1












      1








      1





      $begingroup$


      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







      share|cite|improve this question







      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






      share|cite|improve this question







      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.











      share|cite|improve this question







      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.









      share|cite|improve this question




      share|cite|improve this question






      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

      82




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      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.






















          2 Answers
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          $begingroup$

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






          share|cite|improve this answer









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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
            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



















          2












          $begingroup$

          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}$






          share|cite|improve this answer









          $endgroup$














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            2 Answers
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            active

            oldest

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            2 Answers
            2






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            5












            $begingroup$

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






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              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
















            5












            $begingroup$

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






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              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














            5












            5








            5





            $begingroup$

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






            share|cite|improve this answer









            $endgroup$



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







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 hours 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
              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








            • 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











            2












            $begingroup$

            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}$






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              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}$






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                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}$






                share|cite|improve this answer









                $endgroup$



                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}$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 56 mins ago









                Claude LeiboviciClaude Leibovici

                125k1158135




                125k1158135






















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