I know that BesselY functions defined such that it would isolate the log(x) singularity to only one linearly independent solution so that BesselJ could be well behaved, so the BesselY had to be this way. But when I look at the plots of BesselY, it just looks like a kid haphazardly melded a $\log(x)$ with $\frac{\sin(x+?)}{\sqrt{x}}$. I don't have any particular question, I am just interested to hear if there is anything at all interesting about this.