HallsofIvy was a bit too generous in saying everything is correct.

You cannot evaluate the function at x = 0 and then compute the second set of Cauchy-Riemann equations as you did. This amounts to evaluating a real function f(x,y) at x = 0, computing the partial derivative with respect to y, and then claiming that the result is actually [itex]\frac{\partial f}{\partial y}[/itex].

You must compute the Cauchy-Riemann equations first, then look at the set of (x,y) that satisfy the equations. Then you can determine where the function is analytic (by HallsofIvy's given definition), if anywhere.