I'm dealing with a problem right now that requires me to look at combinations of the form: $$ K_{ i \nu}(x) K_{i \nu}(y) $$
Where $\nu > 0$ is a positive real number, and $x, y \in \mathbb{R}$. Also, $K_{i\nu}$ is the modified Bessel function of the second kind (of order $i \nu$).
I'd like to consider the asymptotic form of the combination $K_{ i \nu}(x) K_{i \nu}(y)$ for two cases: $$ 0 < | x - y | \ll 1 \\ | x - y | \gg 1 $$
Is there any way to do something like this?
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PS. Whittaker and Watson (1965; A Course of Modern Analysis) give the following asysmptotic form for $K_{i \nu}(z)$ for $z$ tending to $0$: $$ K_{i \nu}(z) \ \approx \ \frac{i \pi}{ \sinh( \pi \nu )\ \Gamma(1 + i \nu ) } \left( \frac{z}{2} \right)^{i \nu} \ - \ \frac{i \pi}{ \sinh( \pi \nu )\ \Gamma(1 - i \nu ) } \left( \frac{z}{2} \right)^{-i \nu} $$
Where $\Gamma$ is the ordinary gamma function. But I'm not sure how (if at all) I could utilize this...