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}}</ref><ref>{{Springer|id=B/b015360|title=Bateman function}}</ref> Bateman defined it by

:<math>\displaystyle k_n(x) = \frac{2}{\pi}\int_0^{\pi/2}\cos(x\tan\theta-n\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>

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

The Bateman function 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).</math>

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