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In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From

(1) $$s_{pm} = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right]$$

to

(2) $$s_{pm} = \sinh \left[ \displaystyle \frac{1}{n} \mathrm{arsinh} \left( \frac{1}{\epsilon} \right) \right] \sin (\theta_m) + j \cosh \left[ \frac{1}{n} \mathrm{arsinh} \left( \frac{1}{\epsilon} \right) \right] \cos (\theta_m)$$

where $s$ is the complex variable of the Laplace domain; $m = 1, 2, \ldots, n$ (with $n$ integer) and

$$\theta_m = \displaystyle \frac{\pi}{2} \frac{2m - 1}{n}$$

$\epsilon$ is a real number and $j$ is the imaginary unit.

There are so many expressions for the complex $\arccos$. For example I found that, if

$$w = \arccos (z) = \arccos \displaystyle \frac{j}{\epsilon}$$

with $z$ purely imaginary like in (1), then

$$w = \displaystyle \frac{\pi}{2} + j \mathrm{arsinh} \left( - \displaystyle \frac{1}{\epsilon} \right)$$

or

$$w = \displaystyle \frac{3\pi}{2} + j \mathrm{arsinh} \left( \displaystyle \frac{1}{\epsilon} \right)$$

but I don't know how to use these results to obtain the expression (2) from (1).

In particular, I can't figure out how the terms $\sin (\theta_m)$ and $\cos (\theta_m)$ arise in (2) from $m \pi / n$ in (1).

Which relations should be used and how? Any hint?

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  • $\begingroup$ The argument of the hyperbolic sine $\sinh^{-1}$ is better written $\text{arsinh}$ than $\text{arcsinh}$ (but none are known of LaTeX. $\endgroup$ – Yves Daoust Jun 12 '15 at 9:33
  • $\begingroup$ @YvesDaoust Maybe you are referring not to the argument, but to the function itself: $\mathrm{arsinh}$. You are right and I just edited the question with the correction. $\endgroup$ – BowPark Jun 12 '15 at 9:35
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    $\begingroup$ I am referring to the function "argument of the hyperbolic sine", like $\arccos$ is the function "arc of the cosine". $\endgroup$ – Yves Daoust Jun 12 '15 at 9:38
  • $\begingroup$ @YvesDaoust ok, this is the name you use for the function $\mathrm{arsinh}$. $\endgroup$ – BowPark Jun 12 '15 at 9:48
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    $\begingroup$ Also frequently used notation: $\arg\sinh$, which is easy to render in LaTeX. I've seen in in both Hardy's Course in Pure Mathematics and Spivak's Calculus. $\endgroup$ – Chappers Jun 25 '15 at 10:27

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