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In mathematics, the Bateman function (or k-function) kn is a special case of the confluent hypergeometric function studied by Bateman (1931). Bateman defined it by

$$\displaystyle k_n(x) = \frac{2}{\pi}\int_0^{\pi/2}\cos(x\tan\theta-n\theta) \, d\theta$$

References

Bateman, H. (1931), "The k-function, a particular case of the confluent hypergeometric function", Transactions of the American Mathematical Society 33 (4): 817–831, doi:10.2307/1989510, ISSN 0002-9947, MR1501618
Hazewinkel, Michiel, ed. (2001), "Bateman function", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

Mathematics Encyclopedia