Bessel function of second with complex order

General Usage
4

If you want to use a package that is in development,
add SpecialFunctions and
add ArbNumerics (must use version 0.3.5+, just merged).

using ArbNumerics
import SpecialFunctions: besselj, bessely

for F in (:besselj, :bessely)
   @eval function $F(order::Complex{T}, argument::Complex{T}) where {T}
       result = $F(ArbComplex(order), ArbComplex(argument))
       return Complex{T}(result)
   end
end

then

order = 0.0 + 0.5im; argument = 1.0 + 1.0im;

besselj(order, argument)
# 0.6415000256096106 - 0.2518459064184279im

bessely(order, argument)
# 0.12578491928293956 + 0.17534238733104368im