A cylinder function of order ν is a solution of Bessel’s equation
As before we write D = x(d/dx), so that Bessel’s equation takes the form
For x ∼ 0 this can be viewed as a perturbation of the equation (D2 − ν2)u = 0, which has solutions u(x) = x±ν, so we expect to find solutions that have the form
with f holomorphic near x = 0. Suitably normalized, solutions that have this form are the Bessel ...