ForwardDiff erfcx with complex numbers

ForrwardDiff_erfcx.jl

using ForwardDiff
using SpecialFunctions

import SpecialFunctions.erfcx
function SpecialFunctions.erfcx(ϕ::Complex{<:ForwardDiff.Dual{TAG}}) where {TAG}
  # Split input into real and imaginary parts
  x, y = reim(ϕ)
  # This gives the 'finite' complex part of the input
  z = complex(ForwardDiff.value(x), ForwardDiff.value(y))

  # Calculate the finite part of the output
  q = erfcx(z)
  # split into real and imaginary parts
  u, v = reim(q)

  # split derivative into real and imaginary parts too
  ∂u, ∂v = reim(2 * (z * q - 1 / sqrt(pi)))

  # Now lets deal with the infinitesimals from the real and imaginary parts of ϕ
  px, py = ForwardDiff.partials(x), ForwardDiff.partials(y)
  # Split again into real and imaginary parts
  du = ForwardDiff.Dual{TAG}(u, ∂u*px - ∂v*py)
  dv = ForwardDiff.Dual{TAG}(v, ∂v*px + ∂u*py)

  # And combine
  return complex(du, dv)
end

As per the discourse answer by Mason.