Bateman function

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

:$\displaystyle k_n(x) = \frac\int_0^\cos(x\tan\theta-n\theta) \, d\theta$

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

:$x \frac = (x-n) u$

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

*$k_0(x) = e^$
*$k_(x) = k_n(-x)$
*$k_n(0)=\frac \sin \frac$
*$k_2(x)=(x+, x, ) e^$
*$, k_n(x), \leq 1$ for real values of $n$ and $x$
*$k_(x)=0$ for $x<0$ if $n$ is a positive integer
*k_1(x) = \frac _1(x) + K_0(x) \ x<0, where $K_n(-x)$ is the Modified Bessel function of the second kind.

References

{{Reflist

Special hypergeometric functions
Special functions