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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
 
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
  
<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/s086140/s0861401.png" /></td> </tr></table>
+
$$
 +
H _  \nu  ^ {(} 1) ( z)  =
 +
\frac{1} \pi
 +
\int\limits _ {C _ {1} } e ^ {i z \cos  t }
 +
e ^ {i \nu ( t - \pi / 2) }  dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861402.png" /> is a curve from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861403.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861405.png" />; the Hankel functions of the second kind are given by
+
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
  
<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/s086140/s0861406.png" /></td> </tr></table>
+
$$
 +
H _  \nu  ^ {(} 2) ( z )  =
 +
\frac{1} \pi
 +
\int\limits _ {C _ {2} } e ^ {i z \cos  t }
 +
e ^ {i \nu ( t - \pi /2 ) }  dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861407.png" /> is a curve from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861408.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s0861409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s08614010.png" />; the [[Bessel functions|Bessel functions]] of the first kind are given by
+
where $  C _ {2} $
 +
is a curve from $  \eta - i \infty $
 +
to $  2 \pi - \eta + i \infty $,  
 +
0 \leq  \eta \leq  \pi $;  
 +
the [[Bessel functions|Bessel functions]] of the first kind are given by
  
<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/s086140/s08614011.png" /></td> </tr></table>
+
$$
 +
J _  \nu  ( z )  =
 +
\frac{1}{2 \pi }
 +
\int\limits _ {C _ {3} } e ^ {i z \cos  t }
 +
e ^ {i \nu ( t - \pi / 2 ) }  dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s08614012.png" /> is a curve from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s08614013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s08614014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s08614015.png" />. The representation is valid in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086140/s08614016.png" />, and is named after A. Sommerfeld [[#References|[1]]].
+
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 [[#References|[1]]].
  
 
====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>
 
<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>
 
 
  
 
====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.

Revision as of 08:14, 6 June 2020


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
[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=11611
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article