Bateman 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 functions were first introduced by Havelock in 1927, while investigating the surface elevation of the uniform stream past an immersed circular cylinder. The Havelock function is defined by :\displaystyle h_\nu(x) = \frac{2}{\pi}\int_0^{\pi/2}\sin(x\tan\theta-\nu\theta) \, d\theta . ==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 if n is a positive integer • k_1(x) = \frac{2x}{\pi} [K_1(x) + K_0(x)], \ x, where K_n(-x) is the Modified Bessel function of the second kind ==References==