I have a question about the following statement from the article


* Alves, M., Rivera, J.M., Sepúlveda, M., Villagrán, O.V. and Garay, M.Z., *The asymptotic behavior of the linear transmission problem in viscoelasticity*, Math. Nachr. **287** (2014) pp 483-497, doi:[10.1002/mana.201200319](https://doi.org/10.1002/mana.201200319), ([author pdf](http://www.im.ufrj.br/rivera/Art_Pub/ViscoLocal.pdf))


They state on page 8 that:


Let $\alpha = \dfrac{\rho\_1\lambda^2}{\kappa\_1 + i\kappa\_2\lambda}$ where $\rho\_1, \kappa\_1, \kappa\_2 > 0$. then
$$
\operatorname{Coth}(i\alpha L) \to 1, \ \ \text{ when } \ \ n \to \infty 
$$
where $L > 0$.


Well, firstly, I think they are changing $\alpha$ to $\alpha\_{n}$ and leaving $n$ hidden. And doing some mathematical manipulations, I'm getting to


Let $x\_n = \Re(\alpha\_n L)$ and $y\_n = \Im(\alpha\_n L)$. Using some properties, I can get
$$
\operatorname{Coth}(i\alpha\_n L) = \dfrac{\cos(x\_n)\sin(x\_n)}{\cosh^2 (y\_n) - \cos^2(x\_n)} - \dfrac{\tanh(y\_n)} {\frac{\cos^2(x\_n)}{\cosh^2 (y\_n)} - 1}
$$


From the equality above can I conclude what I want? Why?



$\operatorname{Coth}(\alpha_n a) \to i$ when $n \to \infty $
$\newcommand\la\lambda\newcommand\al\alpha\newcommand\ka\kappa\newcommand\th\theta$First of all, you copied the expression for $\al$ incorrectly. On p. 489 of the paper linked by you (the fourth display from the bottom of the page), we find
$$\al=r(\la)e^{i\th/2},$$
where $\mathbb R\ni\la=\la\_n\to\infty$,
$$r(\la):=\frac{\rho\_1^{1/2}\la}{\sqrt[4]{\ka\_1^2+\ka\_2^2\la^2}}\to\infty,
\quad e^{i\th/2}\to\frac{1-i}{\sqrt2}, 
$$
and $\rho\_1,\ka\_1,\ka\_2$ are fixed positive real numbers.
So,
$$|e^{-2i\al L}|=|\exp(-2iLr(\la)e^{i\th/2})|
=\exp(\Re(-2iLr(\la)e^{i\th/2})) \\ 
=\exp(-(\sqrt2+o(1))\,Lr(\la))\to0.$$
Thus,
$$\coth(i\al L)=\frac{1+e^{-2i\al L}}{1-e^{-2i\al L}}\to\frac{1+0}{1-0}=1,$$
as desired.


