Bateman function

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

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

Bateman discovered this function, when Theodore von Kármán asked for the solution of the following differential equation

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

and Bateman found this function as one of the solution.

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

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]