Bateman function: Difference between revisions

Revision as of 14:50, 16 December 2023

Revision as of 15:04, 16 December 2023

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

==Havelock function==

Complementary to the Bateman function, one may also define the Havelock function, named after [[Thomas Henry Havelock]]. In fact, both the Bateman and the Havelock function was first introduced by Havelock in 1927,<ref>Havelock, T. H. (1927). The method of images in some problems of surface waves. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(771), 268-280.</ref> while investigating the surface elevation of the uniform stream past an immersed circular cylinder. The Havelock function is defined by

:<math>\displaystyle h_\nu(x) = \frac{2}{\pi}\int_0^{\pi/2}\sin(x\tan\theta-\nu\theta) \, d\theta .</math>

==Properties==

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*<math>|k_n(x)|\leq 1</math> for real values of <math>n</math> and <math>x</math>

*<math>k_{2n}(x)=0</math> for <math>x<0</math> if <math>n</math> is a positive integer

*<math>k_1(x) = \frac{2x}{\pi} [K_1(x) + K_0(x)], \ x<0</math>, where <math>K_n(-x)</math> is the [[Modified Bessel function of the second kind]].

==References==

{{Reflist|30em}}