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More commonly called Bessel function, or [[Cylinder functions|Cylinder function]], of the second kind. They were introduced by Neumann in 1867 (and hence the terminology Neumann functions used by some authors). The Bessel functions of the second type $Y_\nu$ (occasionally the notation $N_\nu$ is also used) can be defined in terms of the [[Bessel functions|Bessel functions]] of the first kind $J_\nu$ as follows:
\begin{equation}\label{e:Bessel_f2}
Y_\nu (z) = \frac{J_\nu (z) \cos \nu \pi - J_{-\nu} (z)}{\sin \nu \pi} \qquad \mbox{for}\; \nu\not\in \mathbb Z
\end{equation}
\[
Y_n (z) = \lim_{\nu\to n} Y_\nu (z) \qquad \mbox{for}\; n\in \mathbb Z\, 
\]
where $\nu$ can be any complex number.

When $\nu = p$ real the function $Y_p$ takes real values on the positive real axis and  tends to zero as $x\to\infty$. For large $x$ they have the asymptotic representation
\[
Y_p (x) = \sqrt{\frac{2}{\pi x}} \sin \left(x - \frac{1}{2} p\pi - \frac{\pi}{4}\right) +O\left( \frac{1}{x^{3/2}}\right).
\]
They are connected by the recurrence formulas
\begin{align*}
&Y_{p-1} (x)  + Y_{p+1} (x) = \frac{2p}{x} Y_p (x)\\
&Y_{p-1} (x) - Y_{p+1} (x) = 2 Y'_p (x)\, .
\end{align*}
For integers $p=n$ we have
\[
Y_{-n}  = (-1)^n Y_n\, .
\]
For $p=n$ natural number and small $x$ we have the asymptotic formulas
\[
Y_0 (x) \sim -\frac{2}{\pi} \ln \left(\frac{2}{\gamma x}\right)\, , \qquad Y_n (x) \sim - \frac{(n-1)!}{\pi} \left(\frac{2}{x}\right)^n\, ,
\]
where $\gamma$ is the [[Euler constant]].

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066420a.gif" />

Figure: n066420a

Graphs of some Bessel functions of the second kind.

The Neumann functions of  "half-integral"  order $p = \frac{2n+1}{2}$ can be expressed in terms of the trigonometric functions; in particular,
\[
Y_{1/2} (x) = - \sqrt{\frac{2}{\pi x}} \cos x\, , \qquad Y_{-1/2} (x) = \sqrt{\frac{2}{\pi x}} \sin x\, .
\]

For further references see [[Cylinder functions]].

====References====
* Harold Jeffreys, Bertha Jeffreys, ''Methods of Mathematical Physics'', 3rd edition, Cambridge University Press (1972) Zbl 0238.00004