How to calculate ratio of complex Bessel Functions J0/J2? In order to calculate the frequency-dependent friction in pipe flows, we need to calculate the complex ratio of Bessel functions  $J_0/J_2$, particularly $N(x)=\frac{J_0(x j\sqrt{j})}{J_2(x j\sqrt{j} )}$.
Calculating this naively fails as both $J_0$ and $J_2$ quickly become very large, with standard algorithms (e.g. Matlab) reporting infinity for x>1000.  However, the ratio remains reasonable and slowly, asymptotically approaches a magnitude of 1. 
How can one calculate N for x>1000? 
 A: I understand you are asking for the large argument behaviour of the ratio of two Kelvin functions:
$$\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=J_{\nu}\left(xe^{3\pi i/4%
}\right) $$
Their modulus and phase have rather nice asymptotic expressions for large argument. With the notation
$$\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=M_\nu(x)e^{i\theta_\nu(x)} $$ and $\mu=4\nu^2$ one has
\begin{align}
 M_{\nu}\left(x\right)&=\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\left(1-%
\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}-%
\frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}%
{x^{4}}\right)\right)\\
\theta_{\nu}\left(x\right)&=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}%
\right)\pi+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-%
\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}%
\right)
\end{align}
In the ratio $N(x)$, the diverging exponential term is eliminated and I suppose you can use these asymptotic formula for $x>1000$. The ratio asymptotically approaches -1.
