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The representation of a cylinder function (cf. [[Cylinder functions|Cylinder functions]]) by a contour integral
 
The representation of a cylinder function (cf. [[Cylinder functions|Cylinder functions]]) by a contour integral
  
<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/s/s086/s086160/s0861601.png" /></td> </tr></table>
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$$
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J _  \nu  ( z)  = \
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\frac{1}{2 \pi i }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086160/s0861602.png" /> is arbitrary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086160/s0861603.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086160/s0861604.png" />. Integrals of this type were studied by N.Ya. Sonin (1870).
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\int\limits _ {- \infty } ^ { {( }  0+) }
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e ^ {z ( t  ^ {2} - t ) /2t } t ^ {- \nu - 1 }  dt,
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$$
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where  $  \nu $
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is arbitrary and $  \mathop{\rm Re}  z > 0 $
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or  $  - \pi /2 <  \mathop{\rm arg}  z < \pi /2 $.  
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Integrals of this type were studied by N.Ya. Sonin (1870).
  
 
An integral of the form below is sometimes called a Sonin integral:
 
An integral of the form below is sometimes called a Sonin integral:
  
<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/s/s086/s086160/s0861605.png" /></td> </tr></table>
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$$
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J _ {m + n + 1 }  ( x) =
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$$
  
<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/s/s086/s086160/s0861606.png" /></td> </tr></table>
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$$
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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/s/s086/s086160/s0861607.png" /></td> </tr></table>
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\frac{x ^ {n + 1 } }{2  ^ {n} \Gamma ( n + 1) }
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\int\limits
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_ { 0 } ^ {  \pi  /2 } J _ {m} ( x  \sin  t)  \sin ^
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{m + 1 }  t  \cos ^ {2n + 1 }  t  dt,
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$$
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$$
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\mathop{\rm Re}  m,  \mathop{\rm Re}  n > - 1.
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$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke,  F. Emde,  F. Lösch,  "Tafeln höheren Funktionen" , Teubner  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke,  F. Emde,  F. Lösch,  "Tafeln höheren Funktionen" , Teubner  (1966)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:22, 6 June 2020


The representation of a cylinder function (cf. Cylinder functions) by a contour integral

$$ J _ \nu ( z) = \ \frac{1}{2 \pi i } \int\limits _ {- \infty } ^ { {( } 0+) } e ^ {z ( t ^ {2} - t ) /2t } t ^ {- \nu - 1 } dt, $$

where $ \nu $ is arbitrary and $ \mathop{\rm Re} z > 0 $ or $ - \pi /2 < \mathop{\rm arg} z < \pi /2 $. Integrals of this type were studied by N.Ya. Sonin (1870).

An integral of the form below is sometimes called a Sonin integral:

$$ J _ {m + n + 1 } ( x) = $$

$$ = \ \frac{x ^ {n + 1 } }{2 ^ {n} \Gamma ( n + 1) } \int\limits _ { 0 } ^ { \pi /2 } J _ {m} ( x \sin t) \sin ^ {m + 1 } t \cos ^ {2n + 1 } t dt, $$

$$ \mathop{\rm Re} m, \mathop{\rm Re} n > - 1. $$

References

[1] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2] E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)

Comments

In Western practice one usually writes Sonine integral. Transformed versions of the contour integral were independently obtained by L. Schläfli (1873) and integrals of this type are also called Schläfli integrals. The second mentioned integral is known as Sonine's first finite integral.

References

[a1] G.N. Watson, "The theory of Bessel functions" , 1 , Cambridge Univ. Press (1944) pp. Formulas 6.2(2), 12.11(1)
How to Cite This Entry:
Sonin integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sonin_integral&oldid=11953
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article