Finding asymptotic expansion of Modified Bessel function $I_{\nu}$ and $K_{\nu}$

Question

$$ \frac{I_1\left(\frac{2\sqrt{MN}}{N-1}\right)K_{0}\left(\frac{2\sqrt{M}}{N-1}\right)+I_0\left(\frac{2\sqrt{M}}{N-1}\right)K_{1}\left(\frac{2\sqrt{MN}}{N-1}\right)}{I_1\left(\frac{2\sqrt{MN}}{N-1}\right)K_{1}\left(\frac{2\sqrt{M}}{N-1}\right)-I_1\left(\frac{2\sqrt{M}}{N-1}\right)K_{1}\left(\frac{2\sqrt{MN}}{N-1}\right)}$$ becomes $coth{\sqrt{M}}$, as N approches to 1. I tried with large value approximations for $I_{\nu}$ and $K_{\nu}$, But i am not getting $coth{\sqrt{M}}$.

Answer 1

Keeping the leading term in the asymptotic expansions for $z\to\infty$, \begin{align} I_j(z)&\sim\frac{1}{\sqrt{2\pi}}\frac{e^z}{\sqrt{z}}\\ K_j(z)&\sim\frac{\sqrt{\pi}}{\sqrt{2}}\frac{e^{-z}}{\sqrt{z}} \end{align} for $j=0,1$. Pre-factors of the functions are identical, the ratio is then \begin{equation} \frac{\exp\left( 2\sqrt{M}\frac{\sqrt{N}-1}{N-1} \right)+\exp\left(- 2\sqrt{M}\frac{\sqrt{N}-1}{N-1} \right)}{\exp\left( 2\sqrt{M}\frac{\sqrt{N}-1}{N-1} \right)-\exp\left(- 2\sqrt{M}\frac{\sqrt{N}-1}{N-1} \right)}=\coth\frac{ 2\sqrt{M}}{\sqrt{N}+1} \end{equation} It approaches $\coth \sqrt{M}$ as $N\to 1$.