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The function
 
The function
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\[
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{\bf E}_\nu (x) = \frac{1}{\pi} \int_0^\pi \sin\, (\nu \theta - x \sin \theta)\, d\theta
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\]
 +
where $x$ is a complex variable and $\nu$ a complex parameter, first studied by Weber in {{Cite|We}}. The term Weber function
 +
is also used sometimes for the [[Cylinder functions|cylinder function]] of second type $Y_\nu$ (also called Bessel functions of second type or Neumann functions and denoted sometimes by $N_\nu$).
  
<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/w/w097/w097320/w0973201.png" /></td> </tr></table>
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The Weber function satisfies the inhomogeneous [[Bessel equation|Bessel equation]]
 
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\[
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w0973202.png" /> is a complex number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w0973203.png" /> is a real number. It satisfies the inhomogeneous [[Bessel equation|Bessel equation]]
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x^2 y'' + xy' + (x^2 - \nu^2) y = - \frac{1}{\pi} \left( x+ \nu + (x-\nu) \cos \nu\pi\right)\,  
 
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\]
<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/w/w097/w097320/w0973204.png" /></td> </tr></table>
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(see 10.12 in {{Cite|Wa}}).
 
 
For non-integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w0973205.png" /> the following expansion is valid:
 
 
 
<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/w/w097/w097320/w0973206.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/w/w097/w097320/w0973207.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/w/w097/w097320/w0973208.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w0973209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w09732010.png" />, the following asymptotic expansion is valid:
 
 
 
<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/w/w097/w097320/w09732011.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/w/w097/w097320/w09732012.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/w/w097/w097320/w09732013.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w09732014.png" /> is the [[Neumann function|Neumann function]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w09732015.png" /> is not an integer, the Weber function is related to the [[Anger function|Anger function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097320/w09732016.png" /> 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/w/w097/w097320/w09732017.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/w/w097/w097320/w09732018.png" /></td> </tr></table>
 
  
The Weber functions were first studied by H. Weber [[#References|[1]]].
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When the parameter $\nu$ is not an integer, the Weber function has the following expansion:
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\[
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{\bf E}_\nu (x) = \frac{1-\cos \nu\pi}{\nu \pi} \left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right]
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- \frac{1-\cos \nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right]
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\]
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If $|x|$ is large and $|{\rm arg}\, x|< \frac{\pi}{2}$, the following asymptotic expansion is valid:
 +
\[
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{\bf E_\nu} (x) = - Y_\nu (x) - \frac{1+\cos \nu\pi}{\pi x} \left[1 - \frac{1-\nu^2}{x^2} +
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\frac{(1-\nu^2)(3^2-\nu^2)}{x^4} + \ldots \right] - \nu \frac{1-\cos \nu\pi}{\pi x} \left[
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\frac{1}{x} - \frac{2^2 - \nu^2}{x^3} + \frac{(2^2-\nu^2)(4^2-\nu^2)}{x^5} + \ldots \right]\, .
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\]
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where $Y_\nu$ is the [[Neumann function|Neumann function]]. If $\nu$ is not an integer, the Weber function is related to the [[Anger function|Anger function]] ${\bf J}_\nu$ by the following equations:
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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)\\
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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}
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(cf. 10.11 in {{Cite|Wa}}).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.F. Weber''Zurich Vierteljahresschrift'' , '''24'''  (1879pp. 33–76</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.N. Watson"A treatise on the theory of Bessel functions" , '''1''' , Cambridge Univ. Press (1952)</TD></TR></table>
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{|
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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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|valign="top"|{{Ref|We}}||valign="top"| H.F. Weber''Zurich Vierteljahresschrift'' , '''24'''  (1879) pp. 33–76
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|-
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|}

Latest revision as of 09:12, 22 February 2014

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

The function \[ {\bf E}_\nu (x) = \frac{1}{\pi} \int_0^\pi \sin\, (\nu \theta - x \sin \theta)\, d\theta \] where $x$ is a complex variable and $\nu$ a complex parameter, first studied by Weber in [We]. The term Weber function is also used sometimes for the cylinder function of second type $Y_\nu$ (also called Bessel functions of second type or Neumann functions and denoted sometimes by $N_\nu$).

The Weber function satisfies the inhomogeneous Bessel equation \[ x^2 y'' + xy' + (x^2 - \nu^2) y = - \frac{1}{\pi} \left( x+ \nu + (x-\nu) \cos \nu\pi\right)\, \] (see 10.12 in [Wa]).

When the parameter $\nu$ is not an integer, the Weber function has the following expansion: \[ {\bf E}_\nu (x) = \frac{1-\cos \nu\pi}{\nu \pi} \left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right] - \frac{1-\cos \nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right] \] If $|x|$ is large and $|{\rm arg}\, x|< \frac{\pi}{2}$, the following asymptotic expansion is valid: \[ {\bf E_\nu} (x) = - Y_\nu (x) - \frac{1+\cos \nu\pi}{\pi x} \left[1 - \frac{1-\nu^2}{x^2} + \frac{(1-\nu^2)(3^2-\nu^2)}{x^4} + \ldots \right] - \nu \frac{1-\cos \nu\pi}{\pi x} \left[ \frac{1}{x} - \frac{2^2 - \nu^2}{x^3} + \frac{(2^2-\nu^2)(4^2-\nu^2)}{x^5} + \ldots \right]\, . \] where $Y_\nu$ is the Neumann function. If $\nu$ is not an integer, the Weber function is related to the Anger function ${\bf J}_\nu$ 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

[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
[We] H.F. Weber, Zurich Vierteljahresschrift , 24 (1879) pp. 33–76
How to Cite This Entry:
Weber function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weber_function&oldid=13892
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article