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{{MSC|34}}
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The function
 
The function
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\begin{equation}\label{e:Anger}
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{\bf J}_\nu (x) = \frac{1}{\pi} \int_0^\pi \cos\, (\nu \theta - x \sin \theta)\, d\theta
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\end{equation}
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where  $x$ is a complex variable and $\nu$ a complex parameter. The functions are named after C. T. Anger who in {{Cite|An}} studied the integral on the right hand side of \eqref{e:Anger} when the upper limit is $2\pi$ rather than $\pi$ . The Anger function satisfies the inhomogeneous  [[Bessel equation|Bessel equation]]
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\[
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x^2 y'' + xy' + (x^2 - \nu^2) y = \frac{(z-\nu) \sin \nu\pi}{\pi} \,
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\]
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(see 10.12 in {{Cite|Wa}}).
  
<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/a/a012/a012490/a0124901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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For integers $\nu =n$ the Anger function coincides with the Bessel function $J_\nu$ of order $n$ (cf. [[Bessel functions]]). For non-integer $\nu$ the following expansion is valid:
 
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\[
which satisfies the inhomogeneous Bessel equation:
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{\bf J}_\nu (x) = \frac{\sin \nu\pi}{\nu \pi} \left[ 1 -  \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots  \right]
 
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+ \frac{\sin \nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right]
<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/a/a012/a012490/a0124902.png" /></td> </tr></table>
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\]
 
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For $|x|$ large and $|{\rm arg}\, x| < \pi$ we moreover have the asymptotic expansion
For integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012490/a0124903.png" /> is the Bessel function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012490/a0124904.png" /> (cf. [[Bessel functions|Bessel functions]]). For non-integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012490/a0124905.png" /> the following expansion is valid:
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\[
 
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{\bf J_\nu} (x) = - J_\nu (x) + \frac{\sin \nu\pi}{\pi x} \left[1 - \frac{1-\nu^2}{x^2} +
<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/a/a012/a012490/a0124906.png" /></td> </tr></table>
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\frac{(1-\nu^2)(3^2-\nu^2)}{x^4} + \ldots \right] - \nu \frac{\sin \nu \pi}{\pi x} \left[
 
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\frac{1}{x} - \frac{2^2 - \nu^2}{x^3} + \frac{(2^2-\nu^2)(4^2-\nu^2)}{x^5} + \ldots \right]\, .
<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/a/a012/a012490/a0124907.png" /></td> </tr></table>
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\]
 
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If $\nu$ is not an integer, the Anger function is related to the  [[Weber function]] ${\bf E}_\nu$ by the following  equations:
<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/a/a012/a012490/a0124908.png" /></td> </tr></table>
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\begin{align}
 
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& \sin \nu\pi\, {\bf J}_\nu (x) = \cos \nu \pi\, {\bf E}_\nu (x) - {\bf E}_{-\nu} (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/a/a012/a012490/a0124909.png" /></td> </tr></table>
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& \sin \nu\pi\, {\bf E}_\nu (x) = {\bf J}_{-\nu} (x) - \cos \nu\pi\, {\bf J}_\nu (x)\,  
 
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\end{align}
The asymptotic expansion
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(cf. 10.11 in {{Cite|Wa}}).
 
 
<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/a/a012/a012490/a01249010.png" /></td> </tr></table>
 
 
 
<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/a/a012/a012490/a01249011.png" /></td> </tr></table>
 
 
 
<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/a/a012/a012490/a01249012.png" /></td> </tr></table>
 
 
 
is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012490/a01249013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012490/a01249014.png" />.
 
 
 
The functions have been named after C.T. Anger [[#References|[1]]], who studied functions of the type (*), but with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012490/a01249015.png" /> as the upper limit of the integral.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"C.T. Anger,  ''Neueste Schr. d. Naturf. d. Ges. i. Danzig'' , '''5'''  (1855)  pp. 1–29</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|An}}||valign="top"| C.T. Anger,  ''Neueste Schr. d. Naturf. d. Ges. i. Danzig'' , '''5'''  (1855)  pp. 1–29
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|-
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|valign="top"|{{Ref|Wa}}||valign="top"| G.N. Watson,  "A   treatise on the theory of Bessel functions", '''1–2''', Cambridge Univ.   Press  (1952) {{MR|1349110}} {{MR|1570252}}  {{MR|0010746}}  {{MR|1520278}}    {{ZBL|0849.33001}}  {{ZBL|0174.36202}}  {{ZBL|0063.08184}}
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|-
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Latest revision as of 09:02, 22 February 2014

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

The function \begin{equation}\label{e:Anger} {\bf J}_\nu (x) = \frac{1}{\pi} \int_0^\pi \cos\, (\nu \theta - x \sin \theta)\, d\theta \end{equation} where $x$ is a complex variable and $\nu$ a complex parameter. The functions are named after C. T. Anger who in [An] studied the integral on the right hand side of \eqref{e:Anger} when the upper limit is $2\pi$ rather than $\pi$ . The Anger function satisfies the inhomogeneous Bessel equation \[ x^2 y'' + xy' + (x^2 - \nu^2) y = \frac{(z-\nu) \sin \nu\pi}{\pi} \, \] (see 10.12 in [Wa]).

For integers $\nu =n$ the Anger function coincides with the Bessel function $J_\nu$ of order $n$ (cf. Bessel functions). For non-integer $\nu$ the following expansion is valid: \[ {\bf J}_\nu (x) = \frac{\sin \nu\pi}{\nu \pi} \left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right] + \frac{\sin \nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right] \] For $|x|$ large and $|{\rm arg}\, x| < \pi$ we moreover have the asymptotic expansion \[ {\bf J_\nu} (x) = - J_\nu (x) + \frac{\sin \nu\pi}{\pi x} \left[1 - \frac{1-\nu^2}{x^2} + \frac{(1-\nu^2)(3^2-\nu^2)}{x^4} + \ldots \right] - \nu \frac{\sin \nu \pi}{\pi x} \left[ \frac{1}{x} - \frac{2^2 - \nu^2}{x^3} + \frac{(2^2-\nu^2)(4^2-\nu^2)}{x^5} + \ldots \right]\, . \] If $\nu$ is not an integer, the Anger function is related to the Weber function ${\bf E}_\nu$ by the following equations: \begin{align} & \sin \nu\pi\, {\bf J}_\nu (x) = \cos \nu \pi\, {\bf E}_\nu (x) - {\bf E}_{-\nu} (x)\\ & \sin \nu\pi\, {\bf E}_\nu (x) = {\bf J}_{-\nu} (x) - \cos \nu\pi\, {\bf J}_\nu (x)\, \end{align} (cf. 10.11 in [Wa]).

References

[An] C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig , 5 (1855) pp. 1–29
[Wa] G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184
How to Cite This Entry:
Anger function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anger_function&oldid=31330
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article