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>{{harvtxtCitation | 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_nk_\nu(x) = \frac{2}{\pi}\int_0^{\pi/2}\cos(x\tan\theta-n\nu\theta) \, d\theta .</math>

[[Harry Bateman|Bateman]] discovered this function, when [[Theodore von Kármán]] asked for the solution of the following differential equation which appeared in the theory of [[turbulence]]<ref>Martin, P. A., & Bateman, H. (2010). from Manchester to Manuscript Project. Mathematics Today, 46, 82-85. http://www.math.ust.hk/~machiang/papers_folder/http___www.ima.org.uk_mathematics_mt_april10_harry_bateman_from_manchester_to_manuscript_project.pdf</ref>

:<math>x \frac{d^2u}{dx^2} = (x-n\nu) u</math>

and Bateman found this function as one of the solutionsolutions. Bateman denoted this function as "k" function in honor of [[Theodore von Kármán]].

The Bateman function for <math>x>0</math> is the related to the [[Confluent hypergeometric function]] of the second kind as follows

:<math>k_{\nu}(x)=\frac{e^{-x}}{\Gamma\left(1+\frac{1}{2}\nu\right)} U\left(-\frac{1}{2}\nu,0,2x\right), \quad x>0.</math>

This is not to be confused with another function of the same name which is used in Pharmacokinetics.

==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 functions were 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>

==Properties==

*<math>k_ok_0(x) = e^{-|x|}</math>

*<math>k_{-n}(x) = k_n(-x)</math>

*<math>k_n(0)=\frac{2}{n\pi} \sin \frac{n\pi}{2}</math>

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*<math>|k_n(x)|\leq 1</math> for real values of <math>n</math> and <math>x</math>

*<math>k_{2n}(x)=0</math> for <math>x<0</math> if <math>n</math> is a positive integer

* If <math>n</math> is an odd integer, then <math>k_nk_1(x) = -\frac{2x}{\pi} [K_1(-x) + K_0(-x)], \ x<0</math>, where <math>K_n(-x)</math> is the [[Modified Bessel function of the second kind]].

==References==

{{Reflist|30em}}

*{{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}}

*{{Springer|id=B/b015360|title=Bateman function}}

[[Category:Special hypergeometric functions]]