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'' $  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 [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman,   "The $k$-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>