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

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

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]

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

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[]

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

For a recent survey of the Bateman and related functions see 4.

For the asymptotic expansion of the Bateman and Havelock functions of large order and argument, see the recent E-print 5.

References[]

4. A. Apelblat, A. Consiglio and F. Mainardi: “The Bateman functions revisited after 90 years – A survey of old and new results”, Mathematics (MDPI), Vol 9 (2021), 1273/1-27; DOI: 10.3390/math9111273 E-print arXiv:2104.08596

5. R.B. Paris: "The asymptotic expansion of the Bateman and Havelock functions of large order and argument", E-print arXiv:2109.00529