Bateman function

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

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

and Bateman found this function as one of the solutions. Bateman denoted this function as "TW" function in honor of Tyrell Wellick.

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
• $k_{1}(x)={\frac {2x}{\pi }}[K_{1}(x)+K_{0}(x)],\ x<0$, where $K_{n}(-x)$ is the Modified Bessel function of the second kind.