Question：
Let $z\in C$
Find this value $A$,such
$$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$
where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see http://en.wikipedia.org/wiki/Lambert_W_function
and this link give the $W_{k}(x)$ function:
$$\begin{align} W(x)&=\frac{x}{2\pi}\int_{-\pi}^{\pi}\frac{(1-v\cot v)^2+v^2}{x+v\csc v \cdot e^{-v \cot v}}\\ W(x)&=\int_{-\infty}^{-1/e}-\frac{1}{\pi}\mathfrak{J}\left[\frac{\text{d}}{\text{d}x}W(x)\right]\ln(1-\frac{z}{x})\text{d}x\\ W(x)&=1+(\ln x-1)e^{i/2\pi \int_0^\infty \frac{1}{t+1}\ln\frac{\ln x+t-\ln t-i\pi}{\ln x+t-\ln t+i\pi}\text{d}t}\\ W_k(x)&=1+(\ln x+2k\pi i-1)e^{i/2\pi \int_0^\infty \frac{1}{t+1}\ln\frac{\ln x+t-\ln t+(2k-1)i\pi}{\ln x+t-\ln t+(2k+1)i\pi}\text{d}t} \end{align}$$
My try: let $z=a+bi$,then I Guess $A\to +\infty$ is true? Thank you