Bateman function

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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 k ν ( x ) = 2 π ∫ 0 π / 2 cos ⁡ ( x tan ⁡ θ − ν θ ) d θ . {\displaystyle \displaystyle k_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -\nu \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 x d 2 u d x 2 = ( x − ν ) u {\displaystyle x{\frac {d^{2}u}{dx^{2}}}=(x-\nu )u} and Bateman found this function as one of the solutions. Bateman denoted this function as "k" function in honor of Theodore von Kármán. The Bateman function for x > 0 {\displaystyle x>0} is the related to the Confluent hypergeometric function of the second kind as follows k ν ( x ) = e − x Γ ( 1 + 1 2 ν ) U ( − 1 2 ν , 0 , 2 x ) , x > 0. {\displaystyle k_{\nu }(x)={\frac {e^{-x}}{\Gamma \left(1+{\frac {1}{2}}\nu \right)}}U\left(-{\frac {1}{2}}\nu ,0,2x\right),\quad x>0.} This is not to be confused with another function of the same name which is used in Pharmacokinetics.