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.
[Math] Is the function $f(z)= \sqrt{xy}$ analytic?
analytic-functions
analytic-functions
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.
Best Answer
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 $0x+0y$.