Bateman function: Difference between revisions

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In mathematics, the '''Bateman function''' (or ''k''-function) is a special case of the [[confluent hypergeometric function]] studied by [[Harry Bateman]](1931).<ref>{{Citation | last1=Bateman | first1=H. | authorlink=Harry Bateman | title=The k-function, a particular case of the confluent hypergeometric function | doi=10.2307/1989510 | mr=1501618 | year=1931 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=33 | issue=4 | pages=817–831| jstor=1989510 }}</ref><ref>{{Springer|id=B/b015360|title=Bateman function}}</ref> Bateman defined it by

:<math>\displaystyle k_\nu(x) = \frac{2}{\pi}\int_0^{\pi/2}\cos(x\tan\theta-\nu\theta) \, d\theta .</math>

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==Havelock function==

Complementary to the Bateman function, one may also define the Havelock function, named after [[Thomas Henry Havelock]]. In fact, both the Bateman and the Havelock functionfunctions waswere first introduced by Havelock in 1927,<ref>Havelock, T. H. (1927). The method of images in some problems of surface waves. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(771), 268-280.</ref> while investigating the surface elevation of the uniform stream past an immersed circular cylinder. The Havelock function is defined by

:<math>\displaystyle h_\nu(x) = \frac{2}{\pi}\int_0^{\pi/2}\sin(x\tan\theta-\nu\theta) \, d\theta .</math>