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

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

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

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

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 for ${\displaystyle x>0}$ is the related to the Confluent hypergeometric function of the second kind as follows

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

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