Bessel function ratio approximation Can we say anything about the ratio:
$$\frac{K_1(z)}{K_0(z)}?$$ In particular, can we describe its behaviour for small or large $z\in\mathbb{R}$.
 A: Using the "smoothed" integral representation (i.e., just integrate in [0,L] and then we will take the limit L tending to infinity to conclude), we get that the leading order (in epsilon) is
$$
\frac{K_0}{K_1}(\epsilon)\approx\frac{\int_0^L dt}{\int_0^L \cosh(t)dt}=\frac{L}{\sinh(L)}\rightarrow 0\;\text{ if }L\rightarrow\infty.
$$
In other words, your quotient diverges for small numbers, I think. For big numbers seems harder for me. Maybe using $K_0'=-K_1$ you can figure out something. I don't know.
A: According to http://www.wolframalpha.com/input/?i=lim%28z+to%2Binf%29%28K_1%28z%29%29%2F%28K_0%28z%29%29 and http://www.wolframalpha.com/input/?i=lim%28z+to-inf%29%28K_1%28z%29%29%2F%28K_0%28z%29%29 , $\lim\limits_{z\to\pm\infty}\dfrac{K_1(z)}{K_0(z)}=1$ .
A: We can analyse the asymptotic behaviour of the ratio:
$K_1(z)/K_0(z)$
as z becomes small or large by observing the leading order term of its series expansion. We then have:
$$\frac{K_{1}\left(z\right)}{K_{0}\left(z\right)}\sim\left\{ \begin{array}{cc}
-\frac{1}{z\log z}, & z>0\;\mathrm{small}\\
1, & z\;\mathrm{large}
\end{array}\right..$$
