Bessel function of the second kind

Remarks

The function $Y_\nu$ solves Bessel's equation $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$, because it is a linear combination of solutions for non-integer $\nu$ and a locally uniform limit of solutions for integer $\nu$. Its Wronskian with $J_\nu$ is $W(J_\nu, Y_\nu)(z) = \frac{2}{\pi z}$, which never vanishes, so $J_\nu$ and $Y_\nu$ are independent for every $\nu$, integers included. This is the point of the definition: for an integer $n$ one has $J_{-n} = (-1)^n J_n$, so $J_n$ and $J_{-n}$ do not form a basis, while $J_n$ and $Y_n$ do. For $\nu = \frac{1}{2}$ the formula gives $Y_{1/2}(x) = -J_{-1/2}(x) = -\sqrt{\frac{2}{\pi x}} \cos x$ for $x > 0$. A non-example: the function $J_{-\nu}$ alone is not $Y_\nu$; it is a second solution only when $\nu \notin \Z$. The notation $N_\nu$ is also used for $Y_\nu$.