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Difference between revisions of "Bourget function"

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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017450/b0174501.png" /> which may be defined as a generalization of the integral representation of the [[Bessel functions|Bessel functions]]
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The function $J_{n,k}(z)$ which may be defined as a generalization of the integral representation of the [[Bessel functions|Bessel functions]]
  
<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/b/b017/b017450/b0174502.png" /></td> </tr></table>
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$$J_{n,k}(z)=\frac{1}{2\pi i}\int t^{-n-1}\left(t+\frac1t\right)^k\exp\left\lbrace\frac12z\left(t-\frac1t\right)\right\rbrace dt,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017450/b0174503.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017450/b0174504.png" /> is a positive integer. The integration contour makes one counter-clockwise turn around the coordinate origin. In other words,
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where $n$ is an integer and $k$ is a positive integer. The integration contour makes one counter-clockwise turn around the coordinate origin. In other words,
  
<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/b/b017/b017450/b0174505.png" /></td> </tr></table>
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$$J_{n,k}(z)=\frac1\pi\int\limits_0^\pi(2\cos\theta)^k\cos(n\theta-z\sin\theta)d\theta,$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017450/b0174506.png" /> is a cylinder function of the first kind. So named after J. Bourget [[#References|[1]]], who studied the function with a view to various applications in astronomy.
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$J_{n,0}(z)\equiv J_n(z)$ is a cylinder function of the first kind. So named after J. Bourget [[#References|[1]]], who studied the function with a view to various applications in astronomy.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Bourget,  "Mémoire sur les nombres de Cauchy et leur application à divers problèmes de mécanique céleste"  ''J. Math. Pures Appl. (2)'' , '''6'''  (1861)  pp. 32–54</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)  pp. Chapt. 10</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Bourget,  "Mémoire sur les nombres de Cauchy et leur application à divers problèmes de mécanique céleste"  ''J. Math. Pures Appl. (2)'' , '''6'''  (1861)  pp. 32–54</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)  pp. Chapt. 10</TD></TR></table>

Latest revision as of 12:15, 13 August 2014

The function $J_{n,k}(z)$ which may be defined as a generalization of the integral representation of the Bessel functions

$$J_{n,k}(z)=\frac{1}{2\pi i}\int t^{-n-1}\left(t+\frac1t\right)^k\exp\left\lbrace\frac12z\left(t-\frac1t\right)\right\rbrace dt,$$

where $n$ is an integer and $k$ is a positive integer. The integration contour makes one counter-clockwise turn around the coordinate origin. In other words,

$$J_{n,k}(z)=\frac1\pi\int\limits_0^\pi(2\cos\theta)^k\cos(n\theta-z\sin\theta)d\theta,$$

$J_{n,0}(z)\equiv J_n(z)$ is a cylinder function of the first kind. So named after J. Bourget [1], who studied the function with a view to various applications in astronomy.

References

[1] J. Bourget, "Mémoire sur les nombres de Cauchy et leur application à divers problèmes de mécanique céleste" J. Math. Pures Appl. (2) , 6 (1861) pp. 32–54
[2] G.N. Watson, "A treatise on the theory of Bessel functions" , 1 , Cambridge Univ. Press (1952) pp. Chapt. 10
How to Cite This Entry:
Bourget function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bourget_function&oldid=32896
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article