Bateman function: Difference between revisions

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In mathematics, the '''Bateman function''' (or ''k''-function) is a special case of the [[confluent hypergeometric function]] studied by {{harvtxt|Bateman|1931}}. Bateman defined it by

:<math>\displaystyle k_\nuk_n(x) = \frac{2}{\pi}\int_0^{\pi/2}\cos(x\tan\theta-\nun\theta) \, d\theta</math>

[[Harry Bateman|Bateman]] discovered this function, when [[Theodore von Kármán]] asked for the solution of the following differential equation which appeared in the theory of [[turbulence]]<ref>Martin, 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</ref>

:<math>x \frac{d^2u}{dx^2} = (x-\nun) u</math>

and Bateman found this function as one of the solution. Bateman denoted this function as "k" function in honor of [[Theodore von Kármán]].