Namespaces
Variants
Actions

Difference between revisions of "Neumann function"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
m (made the asymptotics precise)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
{{MSC|33C10}}
 
{{MSC|33C10}}
{{TEX|part}}
+
{{TEX|done}}
  
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:
+
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}
 
\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
 
Y_\nu (z) = \frac{J_\nu (z) \cos \nu \pi - J_{-\nu} (z)}{\sin \nu \pi} \qquad \mbox{for}\; \nu\not\in \mathbb Z
Line 13: Line 13:
 
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
 
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)
+
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
 
They are connected by the recurrence formulas
Line 41: Line 41:
 
\]
 
\]
  
For further references see [[Cylinder function]].
+
For further references see [[Cylinder functions]].
 +
 
 +
====References====
 +
* Harold Jeffreys, Bertha Jeffreys, ''Methods of Mathematical Physics'', 3rd edition, Cambridge University Press (1972) Zbl 0238.00004

Latest revision as of 11:28, 14 June 2019

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 \[ Y_p (x) = \sqrt{\frac{2}{\pi x}} \sin \left(x - \frac{1}{2} p\pi - \frac{\pi}{4}\right) +O\left( \frac{1}{x^{3/2}}\right). \] 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.

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
How to Cite This Entry:
Neumann function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_function&oldid=31407
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article