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A cylinder function (cf. [[Cylinder functions|Cylinder functions]]) of the second kind. The Neumann functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664201.png" /> (occasionally the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664202.png" /> is used) can be defined in terms of the [[Bessel functions|Bessel functions]] as follows:
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{{MSC|33C10}}
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{{TEX|part}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664203.png" /></td> </tr></table>
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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:
 
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\begin{equation}\label{e:Bessel_f2}
They are real for positive real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664204.png" /> and tend to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664205.png" />. For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664206.png" /> they have the asymptotic representation
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Y_\nu (z) = \frac{J_\nu (z) \cos \nu \pi - J_{-\nu} (z)}{\sin \nu \pi} \qquad \mbox{for}\; \nu\not\in \mathbb Z
 
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\end{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664207.png" /></td> </tr></table>
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\[
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Y_n (z) = \lim_{\nu\to n} Y_\nu (z) \qquad \mbox{for}\; n\in \mathbb Z\,
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\]
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where $\nu$ can be any complex number.
  
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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
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\[
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Y_p (x) = \sqrt{\frac{2}{\pi x} \sin \left(x - \frac{1}{2} p\pi - \frac{\pi}{4}\right)
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\]
 
They are connected by the recurrence formulas
 
They are connected by the recurrence formulas
 
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\begin{align*}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664208.png" /></td> </tr></table>
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&Y_{p-1} (x)  + Y_{p+1} (x) = \frac{2p}{x} Y_p (x)\\
 
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&Y_{p-1} (x) - Y_{p+1} (x) = 2 Y'_p (x)\, .
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n0664209.png" /></td> </tr></table>
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\end{align*}
 
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For integers $p=n$ we have
For integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642010.png" />:
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\[
 
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Y_{-n}  = (-1)^n Y_n\, .
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642011.png" /></td> </tr></table>
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\]
 
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For $p=n$ natural number and small $x$ we have the asymptotic formulas
for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642012.png" />:
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\[
 
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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\, ,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642013.png" /></td> </tr></table>
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\]
 
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where $\gamma$ is the [[Euler constant]].
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642014.png" /> is the Euler constant.
 
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066420a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066420a.gif" />
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Figure: n066420a
 
Figure: n066420a
  
Graphs of Neumann functions.
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Graphs of some Bessel functions of the second kind.
 
 
The Neumann functions of  "half-integral"  order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642015.png" /> can be expressed in terms of the trigonometric functions; in particular,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066420/n06642016.png" /></td> </tr></table>
 
 
 
They were introduced by C.G. Neumann in 1867.
 
  
For references see [[Cylinder functions|Cylinder functions]].
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The Neumann functions of  "half-integral"  order $p = \frac{2n+1}{2}$ can be expressed in terms of the trigonometric functions; in particular,
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\[
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Y_{1/2} (x) = - \sqrt{\frac{2}{\pix}} \cos x\, , \qquad Y_{-1/2} (x) = \sqrt{\frac{2}{\pi x}} \sin x\, .
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\]

Revision as of 17:32, 1 April 2014

2020 Mathematics Subject Classification: Primary: 33C10 [MSN][ZBL]

More commonly called Bessel function, or 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 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 \[ Y_p (x) = \sqrt{\frac{2}{\pi x} \sin \left(x - \frac{1}{2} p\pi - \frac{\pi}{4}\right) \] 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.

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, \[ Y_{1/2} (x) = - \sqrt{\frac{2}{\pix}} \cos x\, , \qquad Y_{-1/2} (x) = \sqrt{\frac{2}{\pi x}} \sin x\, . \]

How to Cite This Entry:
Neumann function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_function&oldid=31406
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article