Complex derivatives in julialang, taking only the first Wirtinger derivative

ComplexDerivatives.jl

using ForwardDiff, StaticArrays

function jacobi(f, x)
  x_real = @SArray [real(x), imag(x)]
  function f_real(v)
    z = complex(v[1], v[2])
    result = f(z)
    return @SArray [real(result), imag(result)]
  end
  J = ForwardDiff.jacobian(f_real, x_real)
  du = complex(J[1,1], J[2,1])
  dv = complex(J[1,2], J[2,2])
  return du, dv
end

function jacobi(f, x)
  x_real = @SArray [real(x), imag(x)]
  function f_real(v)
    z = complex(v[1], v[2])
    result = f(z)
    return @SArray [real(result), imag(result)]
  end
  J = ForwardDiff.jacobian(f_real, x_real)
  du = complex(J[1,1], J[2,1])
  dv = complex(J[1,2], J[2,2])
  return du, dv
end

function wirtinger(f, x)
  du, dv = jacobi(f, x)
  df_dz = (du - im * dv) / 2
  df_dz̄ = (du + im * dv) / 2
  return (dz = df_dz, dz̄ = df_dz̄)
end

complexderivative(f, z) = wirtinger(f, z).dz