Bateman function

In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931)[1][2]. Bateman defined it by

\displaystyle k_{n}(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -n\theta )\,d\theta

Bateman discovered this function, when Theodore von Kármán asked for the solution of the following differential equation which appeared in the theory of turbulence[3]

x{\frac {d^{2}u}{dx^{2}}}=(x-n)u

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

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

Properties

$k_{0}(x)=e^{-|x|}$
$k_{-n}(x)=k_{n}(-x)$
$k_{n}(0)={\frac {2}{n\pi }}\sin {\frac {n\pi }{2}}$
$k_{2}(x)=(x+|x|)e^{-|x|}$
$|k_{n}(x)|\leq 1$ for real values of $n$ and $x$
$k_{2n}(x)=0$ for $x<0$ if $n$ is a positive integer
If $n$ is an odd integer, then $k_{n}(x)=-{\frac {2x}{\pi }}[K_{1}(-x)+K_{0}(-x)],\ x<0$ , where $K_{n}(-x)$ is the Modified Bessel function of the second kind.

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, MR 1501618
"Bateman function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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

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[1] 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, MR 1501618

[2] "Bateman function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

[3] 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