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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b0153602.png" />-function''
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'' $  k $-
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function''
  
 
The function
 
The function
  
<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/b015/b015360/b0153603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
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k _  \nu  (x)  = \
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\frac{2} \pi
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\int\limits _ { 0 } ^ {  \pi  /2 }
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\cos (x  \mathop{\rm tg}  \theta - \nu \theta ) d \theta ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b0153604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b0153605.png" /> are real numbers. The function was defined by H. Bateman [[#References|[1]]]. The Bateman function may be expressed in the form of a confluent hypergeometric function of the second kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b0153606.png" />:
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where $  x $
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and $  \nu $
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are real numbers. The function was defined by H. Bateman [[#References|[1]]]. The Bateman function may be expressed in the form of a confluent hypergeometric function of the second kind $  \Psi (a, b, x) $:
  
<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/b015/b015360/b0153607.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
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\Gamma ( \nu +1)k _ {2 \nu }  (x)  = \
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e  ^ {-x} \Psi ( - \nu , 0 ; 2 x) ,\
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x > 0 .
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$$
  
The relation (2) is conveniently taken as the definition of the Bateman function in the complex plane with the cut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b0153608.png" />. The following relations are valid: for case (1)
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The relation (2) is conveniently taken as the definition of the Bateman function in the complex plane with the cut $  (- \infty , 0] $.  
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The following relations are valid: for case (1)
  
<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/b015/b015360/b0153609.png" /></td> </tr></table>
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$$
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k _  \nu  (-x)  = k _ {- \nu }  (x),
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$$
  
 
for case (2)
 
for case (2)
  
<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/b015/b015360/b01536010.png" /></td> </tr></table>
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$$
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k _ {2 \nu }  (- \xi \pm  i0)  = \
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k _ {- 2 \nu }  ( \xi )-e ^ {\pm  \nu \pi i }
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e  ^  \xi  \Phi (- \nu , 0; - 2 \xi ),
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b01536011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b01536012.png" /> is a confluent hypergeometric function of the first kind.
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where $  \xi > 0 $,  
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and $  \Phi (a, b;  x) $
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is a confluent hypergeometric function of the first kind.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b01536013.png" />-function, a particular case of the confluent hypergeometric function"  ''Trans. Amer. Math. Soc.'' , '''33'''  (1931)  pp. 817–831</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015360/b01536013.png" />-function, a particular case of the confluent hypergeometric function"  ''Trans. Amer. Math. Soc.'' , '''33'''  (1931)  pp. 817–831</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR></table>

Revision as of 10:33, 29 May 2020


$ k $- function

The function

$$ \tag{1 } k _ \nu (x) = \ \frac{2} \pi \int\limits _ { 0 } ^ { \pi /2 } \cos (x \mathop{\rm tg} \theta - \nu \theta ) d \theta , $$

where $ x $ and $ \nu $ are real numbers. The function was defined by H. Bateman [1]. The Bateman function may be expressed in the form of a confluent hypergeometric function of the second kind $ \Psi (a, b, x) $:

$$ \tag{2 } \Gamma ( \nu +1)k _ {2 \nu } (x) = \ e ^ {-x} \Psi ( - \nu , 0 ; 2 x) ,\ x > 0 . $$

The relation (2) is conveniently taken as the definition of the Bateman function in the complex plane with the cut $ (- \infty , 0] $. The following relations are valid: for case (1)

$$ k _ \nu (-x) = k _ {- \nu } (x), $$

for case (2)

$$ k _ {2 \nu } (- \xi \pm i0) = \ k _ {- 2 \nu } ( \xi )-e ^ {\pm \nu \pi i } e ^ \xi \Phi (- \nu , 0; - 2 \xi ), $$

where $ \xi > 0 $, and $ \Phi (a, b; x) $ is a confluent hypergeometric function of the first kind.

References

[1] H. Bateman, "The -function, a particular case of the confluent hypergeometric function" Trans. Amer. Math. Soc. , 33 (1931) pp. 817–831
[2] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
How to Cite This Entry:
Bateman function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bateman_function&oldid=16868
This article was adapted from an original article by L.N. Karmazina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article