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I have the following math symbol in a German thesis written in 1963.

enter image description here

Is it anything more than just a function name?

It is used in the following context

enter image description here

and then goes on to state that "If the expression is larger than 1, then the cosine must be used instead of "the symbol".

enter image description here "Wenn der obige Ausdruck größer als $1$ ist, so muß statt "des Symbols" der cos eingesetzt werden."

The following is my current understanding of the equation given the answers below:

$$ h_{0} \cong 1.15\sqrt{\frac{|G|}{|F|(\frac{e}{\ell} - 1)(1 - \mathcal{X})}}\frac{d^2}{4\ell} \cosh\left(\frac{\alpha}{3}\right) $$ with $$ \cosh\alpha = \frac{|W|}{|G|}\sqrt{\frac{|F|}{|G|}}\frac{Hr_{a}}{T_1 + T_2}\frac{\ell}{d^2}\sqrt{\left(\frac{e}{\ell} - 1\right){\left(1 - \mathcal{X}\right)}} $$

in the case where

$$ \frac{|G|^3}{|F||W|^2}\frac{d^4}{r_a^2l^2}\left(\frac{T_1+T_2}{H} \right)^2 \frac{6\cdot 10^{-5}}{(\frac{e}{l}-1)(1-\mathcal{X})} < 1 $$

when the above equation is greater than 1 then the cosh must be replaced by cos.

Not yet clear where the $6 \cdot 10^{-5}$ is entering from.

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    $\begingroup$ It's "Cos" in "deutsche Schrift". If I remember correctly, some people used $\operatorname{Cos}$ to denote the hyperbolic cosine, $\cosh$, in the old days. $\endgroup$ – Daniel Fischer Aug 31 '14 at 15:30
  • $\begingroup$ Might I enquire as to how this became relevant to you? $\endgroup$ – zibadawa timmy Aug 31 '14 at 15:48
  • $\begingroup$ @zibadawatimmy I am working on book and this equation is in one of the conclusions of a section in a key reference. Want to make sure I am understanding the nuances of the German text correctly. $\endgroup$ – Andrew Aug 31 '14 at 16:07
  • $\begingroup$ @jwpat7 Correction noted. $\endgroup$ – Andrew Aug 31 '14 at 18:52
  • $\begingroup$ it looks like "log" to me $\endgroup$ – pqnet Aug 31 '14 at 18:57
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Note the "C" (third character in third row) and the "s" (fourth character in second row) above.

The important thing is Daniel Fischer's comment from above that "$\operatorname{Cos}$" here does not mean $\cos$ but rather $\cosh$.

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    $\begingroup$ Yes, but "the symbol" $Cos(\alpha)$ does not mean $\cos(\alpha)$, but rather $\cosh(\alpha)$; see the last sentence of the question. $\endgroup$ – Dietrich Burde Aug 31 '14 at 15:35
  • $\begingroup$ @DietrichBurde: I didn't say it meant $\cos$. I posted this after Daniel Fischer's comment above. $\endgroup$ – Frunobulax Aug 31 '14 at 15:37
  • $\begingroup$ The question was already answered. I provided context. What's the problem? $\endgroup$ – Frunobulax Aug 31 '14 at 15:41
  • $\begingroup$ There are lots of other locations in the thesis with "cos" and "sin" in the equations. This is the only location with the "Cos" symbol. $\endgroup$ – Andrew Aug 31 '14 at 15:41
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    $\begingroup$ It is cosinus hyberbolicus, currently it is "cosh" in German. But in older times when the cursive was used, "Cos" with a capital 'C' does indeed mean cosinus hyperbolicus while the "cos" with a small letter means cosinus. $\endgroup$ – Thorsten S. Aug 31 '14 at 16:25
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To guess, except x everything else is a constant.

Left hand side of entire last line is proportional to 1/$\sqrt(1- x)$. So, if

x < 1 the first line is $ cosh( \alpha/3) $ for real argument and, if

x > 1 then first line should be taken to mean $ cos( \alpha/3) $ for imaginary argument

since $ cosh (i \alpha) = cos ( \alpha) $.

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  • $\begingroup$ Here is my current understanding of the equation $\endgroup$ – Andrew Aug 31 '14 at 17:55

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