The Bateman function 
is the special function defined for real 
and 
by the integral
 |
(1)
|
It was studied by Bateman (1931) in connection with an ordinary differential equation arising in the theory of turbulence.
For 
,
it can also be defined by
 |
(2)
|
where 
is the gamma function and 
is the confluent
hypergeometric function of the second kind. At negative even values of 
, the reciprocal gamma function
factor is interpreted by continuity, giving 
for integers 
and 
.
The ordinary differential equation satisfied for 
is
 |
(3)
|
The first few even-order cases for 
are
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
At the origin,
 |
(7)
|
for 
,
with 
(Bateman 1931, Apelblat et al. 2021).
-Function,
a Particular Case of the Confluent Hypergeometric Function." Trans. Amer.
Math. Soc. 33, 817-831, 1931.Koepf, W.