I'm attempting to find a precise as possible approximation to the logarithm of the Modified Bessel Function of the Second Kind:
$$\log K_{\alpha}(x) = \log\Big[\frac{1}{2}\int_0^{∞} t^{\alpha-1} \exp\{-\frac{1}{2}x(t+t^{-1})\}dt\Big].$$
The problem is that $K_{\alpha}(x)$ diverges both as $\alpha\rightarrow\infty$ and as $x\rightarrow0$, which is why I'm taking the logarithm in the first place. But of course the log trick only works if I can use it to break up the expression in some way, which I can't do in this case because I can't pass it through the integral.
There is an asymptotic approximation for large $\alpha$, given by:
$$K_{\alpha}(x)\sim\sqrt{\frac{\pi}{2\alpha}}\Big(\frac{ex}{2\alpha}\Big)^{-\alpha}.$$
Clearly this approximation can be broken up by the log, unfortunately it is not sufficiently precise (for my purposes) for values of $\alpha$ and $x$ which cause overflow.
Does anyone have another way to get a more precise estimate for $\log K_{\alpha}(x)$?
If you want you can assume that $\alpha=k-\frac{1}{2}$ for $k\in{\bf Z}_+$.