Is Lambert W Function in complex domain $(x+iy)e^{x+iy}=a+ib$ solvable? I am solving by method of approximation complex Lambert $W$ Function $(x+iy)e^{x+iy}=a+ib$ for $x$ and $y$, when real values of $a$ and $b$ are given and $i=\sqrt{-1}$. I want to know whether such type of function has been solved earlier. If answer is in affirmative, I pray, I may kindly be informed of its reference or any other source.

$$(x+iy)e^x(\cos(y)+i\sin(y))=a+ib$$
$$\Rightarrow e^x(x\cos(y)-y\sin(y))=a ~~~\text{and} ~~~e^x(y\cos(y)+x\sin(y))=b$$
$$\Rightarrow x=\frac{(a+b)y\cos(y)}{b(\cos(y)-\sin(y))}$$
Plug $x$ into any of above two equations and numerically solve for $y$.
