Bessel function of the second kind ($Y_{\nu }(z)$)

For noninteger order, $Y_{\nu }=(\cos (\pi \nu )J_{\nu }-J_{-\nu })/\sin (\pi \nu )$; the integer-order function is defined by a limit. It gives a second independent solution of the Bessel differential equation. At order zero, $Y_0(z)\sim (2/\pi )[\log (z/2)+\gamma ]$ near positive zero and $Y_0(z)\sim \sqrt{2/(\pi z)}\sin (z-\pi /4)$ at large positive argument, where $\gamma$ is the Euler--Mascheroni constant.

The spherical second-kind function is $n_{\ell }(z)=\sqrt{\pi /(2z)}{Y}_{\ell +1/2}(z)$. Together with the first-kind Spherical Bessel function, it spans the radial free-wave solutions; $n_{\ell }(z)\sim -\cos (z-\ell \pi /2)/z$ at large positive $z$.