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Difference between revisions of "Sommerfeld integral"

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An integral representation of the [[Cylinder functions|cylinder functions]] by a contour integral: The [[Hankel functions|Hankel functions]] of the first kind are given by
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An integral representation of the [[cylinder functions]] by a contour integral: The [[Hankel functions]] of the first kind are given by
  
 
$$  
 
$$  
H _  \nu  ^ {(} 1) ( z)  =   
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H _  \nu  ^ {(1)} ( z)  =   
 
\frac{1} \pi  
 
\frac{1} \pi  
 
  \int\limits _ {C _ {1} } e ^ {i z \cos  t }
 
  \int\limits _ {C _ {1} } e ^ {i z \cos  t }
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$$  
 
$$  
H _  \nu  ^ {(} 2) ( z )  =   
+
H _  \nu  ^ {(2)} ( z )  =   
 
\frac{1} \pi  
 
\frac{1} \pi  
 
  \int\limits _ {C _ {2} } e ^ {i z \cos  t }
 
  \int\limits _ {C _ {2} } e ^ {i z \cos  t }
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Sommerfeld,   "Mathematische Theorie der Diffraction"  ''Math. Ann.'' , '''47'''  (1896)  pp. 317–374</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke,   F. Emde,   "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</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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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Sommerfeld, "Mathematische Theorie der Diffraction"  ''Math. Ann.'' , '''47'''  (1896)  pp. 317–374 {{ZBL|27.0706.03}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  G.N. Watson, "A treatise on the theory of Bessel functions" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR>
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</table>
  
 
====Comments====
 
====Comments====
 
The Hankel functions are also called Bessel functions of the first kind.
 
The Hankel functions are also called Bessel functions of the first kind.

Latest revision as of 17:50, 7 June 2024


An integral representation of the cylinder functions by a contour integral: The Hankel functions of the first kind are given by

$$ H _ \nu ^ {(1)} ( z) = \frac{1} \pi \int\limits _ {C _ {1} } e ^ {i z \cos t } e ^ {i \nu ( t - \pi / 2) } dt , $$

where $ C _ {1} $ is a curve from $ - \eta + i \infty $ to $ \eta - i \infty $, $ 0 \leq \eta \leq \pi $; the Hankel functions of the second kind are given by

$$ H _ \nu ^ {(2)} ( z ) = \frac{1} \pi \int\limits _ {C _ {2} } e ^ {i z \cos t } e ^ {i \nu ( t - \pi /2 ) } dt , $$

where $ C _ {2} $ is a curve from $ \eta - i \infty $ to $ 2 \pi - \eta + i \infty $, $ 0 \leq \eta \leq \pi $; the Bessel functions of the first kind are given by

$$ J _ \nu ( z ) = \frac{1}{2 \pi } \int\limits _ {C _ {3} } e ^ {i z \cos t } e ^ {i \nu ( t - \pi / 2 ) } dt , $$

where $ C _ {3} $ is a curve from $ - \eta + i \infty $ to $ 2 \pi - \eta + i \infty $, $ 0 \leq \eta \leq \pi $. The representation is valid in the domain $ - \eta < \mathop{\rm arg} z < \pi - \eta $, and is named after A. Sommerfeld [1].

References

[1] A. Sommerfeld, "Mathematische Theorie der Diffraction" Math. Ann. , 47 (1896) pp. 317–374 Zbl 27.0706.03
[2] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3] G.N. Watson, "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)

Comments

The Hankel functions are also called Bessel functions of the first kind.

How to Cite This Entry:
Sommerfeld integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sommerfeld_integral&oldid=48749
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article