---
tag: E76951YS23
kind: definition
name: Bessel function of the second kind
msc: [33C10, 34B30]
defines: GRSNDJG363
uses: [CZCMRDWDYW]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel_function
---

## Definition

Let $J_\nu$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$, taken on $\C \setminus (-\infty, 0]$ with the principal branch of $z^\nu$.
For $\nu \in \C \setminus \Z$, the *Bessel function of the second kind* of order $\nu$ is
$$Y_\nu(z) = \frac{J_\nu(z) \cos(\nu \pi) - J_{-\nu}(z)}{\sin(\nu \pi)}.$$
For an integer $n$, it is the limit
$$Y_n(z) = \lim_{\nu \to n} Y_\nu(z).$$

## 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 $Y_\nu$.