Is the function $f(z)= \sqrt{xy}$ analytic??

Question

Whether the function $f(z)= \sqrt{xy}$ is analytic at the origin $(0,0)$ or not?

I want to know how to check/verify using Cauchy-Riemann equations.

Answer 1

If it were analytic (rather than just differentiable) at the origin, it would be complex differentiable in an entire neighborhood of the origin. But it is not even real differentiable at $z=\varepsilon+0i$ for any $\varepsilon>0$ (consider for example $f\circ g$ with $g(t)=\varepsilon+it$).

But it can't even be real differentible at the origin. The partial derivatives are all $0$, but on the line $x=y$ we have $f(x+iy)=x$ -- which is not approximated by $0x+0y$.