MethodError caused by implementation of complex rationals in a small region (function fails for -0.6 + 0.0*im < z < -0.4 + 0.0*im)

[RobbieMackenzie]

Hi,

Thanks for the package, I use it a lot.

I've noticed an issue with the implementation. There is a small region of complex floating-point arguments which trigger this line (line 151) in the case structure:-

w = abs(z+ 1//2) < 1//10 ? imag(z) > 0 ? complex(pointseven, pointseven) : complex(pointseven, -pointseven) : z

If w is assigned either complex(pointseven, pointseven) or complex(pointseven, -pointseven), then the root finding raises a MethodError, because of the function signature (line 168)

function lambertw_root_finding(z::T, x0::T, maxits) where T <: Number

which expects z and x0 to have the same type, but it will get z::ComplexF64 and x0::Complex{Rational{Int64}}.

On the real line this corresponds to approximately -0.6 < z -0.4, and it doesn't take much of an imaginary part before the problematic line isn't triggered, which may explain why no-one's spotted it until now!

Thanks