If W = log_eZ = log_e(x + iy) where i=√-1 , then which of the following is correct?

The correct option is: W is analytic everywhere except at Z = 0

Let us analyze the complex function $W$ defined by:
$W={\log }_{e}Z$
where $Z=x+iy$ is a complex variable and $i=\sqrt{-1}$.

To determine where the function $W=f\left(Z\right)$ is analytic, we examine its derivative with respect to $Z$:
$\frac{dW}{dZ}=\frac{d}{dZ}\left(\log Z\right)=\frac{1}{Z}$

For a function to be analytic at a point, its derivative must exist and be unique in a neighborhood of that point.
Looking at the derivative $\frac{dW}{dZ}=\frac{1}{Z}$, we can see that this derivative is well-defined and exists for all complex numbers $Z$, except when the denominator is zero. That is, the derivative does not exist at:
$Z=0$

At $Z=0$ (which corresponds to $x=0$ and $y=0$), the logarithmic function ${\log }_e\left(0\right)$ is undefined, and the function is not differentiable. Hence, $W$ fails to be analytic at $Z=0$.

For all other points in the complex plane where $Z\ne 0$ (excluding any chosen branch cuts for the multi-valued logarithm to define a single-valued analytic branch), the derivative exists. Therefore, $W$ is analytic everywhere except at $Z=0$.