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Simple Tanh(x) approximation

  1. Jul 24, 2012 #1
    Hello! So I was reading a paper in which I came across the following:

    [itex]
    k = \pi + \pi\ell
    [/itex]

    [itex]
    \tanh{(k)} \approx \pi\ell
    [/itex]

    where "l" is very small. What on earth is the origin of this approximation? I'm sure it's very simple, but I can't seem to derive it from the angle-sum and small angle approximations for sinh and cosh.

    Thanks very much for any help!
     
  2. jcsd
  3. Jul 24, 2012 #2
    I don't know for sure, but I'd try the power series for [itex]\tanh[/itex]. And also [itex]\pi +\pi l \simeq \pi[/itex] or linear apporximation.
     
  4. Jul 24, 2012 #3

    Mentallic

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    Unless I'm missing something we have [tex]\lim_{\ell \to 0}\tanh(k)=\tanh(\pi)\neq 0[/tex] so how could this approximation possibly be true since [tex]\lim_{\ell \to 0}\pi \ell = 0[/tex]
     
  5. Jul 24, 2012 #4
    Yeah, I was just thinking about that - I tried to do a linear approximation and it didn't work. When I first suggested this, I tried to quickly do it in my head, and got it a bit confused and made an error. I'm not sure if OP made a typo.

    If he meant [itex]\tan[/itex] then I think it is right:

    [tex]y = \sec^2(\pi)(x-\pi) + \tan(\pi) = x - \pi[/tex]

    Now if we let [itex]x = \pi +l \pi[/itex] we have the correct linear approximation and [itex]\tan(k) \simeq l \pi[/itex].
     
  6. Jul 24, 2012 #5

    Mentallic

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    Yeah for tan it works. They're both equivalent at [itex]\ell = 0[/itex] and their first derivatives match too.
     
  7. Jul 24, 2012 #6
    OH! So I correctly quoted the paper, but the paper itself was incorrect. The author definitely meant tan()-- he switched between two equations. The previous equations had cosh(.) and sinh(.), so I didn't catch the error:

    Check out 7b in this paper if you're curious where this is from:
    http://www.ctsystemes.com/zeland/publi/00982223.pdf [Broken]

    thanks very much guys!
     
    Last edited by a moderator: May 6, 2017
  8. Jul 24, 2012 #7

    Mentallic

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    At least he plugged the approximation into the formula correctly :wink:
    An honest typo since the Author kept switching between tan and tanh.
     
    Last edited by a moderator: May 6, 2017
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