Is there any result on the asymptotic behaviour of Bessel function of the second kind with a negative order? What I have found is the behaviour when the order $Re(\nu)>0$. For example, it is shown in pp. 360 in [1] that
$$Y_v(z)\approx -(1/\pi)\Gamma(\nu)(\frac12 z)^{-\nu}, Re(\nu)>0.$$
What is the behaviour when $Re(\nu)<0$? Thanks very much.
[1] ABRAMOWITZ M, STEGUN I A 1972. Handbook Of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [M]. Dover Publications; New York.
You find the general asymptotic behavior here. For the case where the order is not an negative integer, we have (valid for small $z$) $$ Y_\alpha(z) \sim-\dfrac{\Gamma(\alpha)}{\pi} \left( \dfrac{2}{z} \right)^\alpha + \dfrac{1}{\Gamma(\alpha+1)} \left(\dfrac{z}{2} \right)^\alpha \cot(\alpha \pi) \,.$$ For $\alpha>0$, the first term dominates and you obtain the result, you have quoted in the question.
For $\alpha<0$, the second term dominates, and we have $$Y_\alpha(z) \sim \dfrac{1}{\Gamma(\alpha+1)} \left(\dfrac{z}{2} \right)^\alpha \cot(\alpha \pi).$$
If $\alpha = -n$ is a negative integer, we have instead $$ Y_{-n}(z) \sim -\dfrac{(-1)^{n}\Gamma(n)}{\pi} \left( \dfrac{2}{z} \right)^{n}\,.$$