Bateman function

$k$- 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 $k$-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)