where and are real numbers. The function was defined by H. Bateman . The Bateman function may be expressed in the form of a confluent hypergeometric function of the second kind :
The relation (2) is conveniently taken as the definition of the Bateman function in the complex plane with the cut . The following relations are valid: for case (1)
for case (2)
where , and is a confluent hypergeometric function of the first kind.
|||H. Bateman, "The -function, a particular case of the confluent hypergeometric function" Trans. Amer. Math. Soc. , 33 (1931) pp. 817–831|
|||H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)|
Bateman function. L.N. Karmazina (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bateman_function&oldid=16868