The cylinder function is defined as
(1)

The Bessel functions are sometimes also called cylinder functions.
To find the Fourier transform of the cylinder function, let
(2)
 
(3)

and
(4)
 
(5)

Then
(6)
 
(7)
 
(8)

Let , so . Then
(9)
 
(10)
 
(11)
 
(12)
 
(13)

where is a Bessel function of the first kind.
As defined by Watson (1966), a "cylinder function" is any function which satisfies the recurrence relations
(14)

(15)

This class of functions can be expressed in terms of Bessel functions.