Consider the functions $f(z)=x^2+iy^2$ and $g(z)=x^2+y^2+ixy$. At $z=0$,

Question

Consider the functions $f(z)=x^2+iy^2$ and $g(z)=x^2+y^2+ixy$. At $z=0$,

(a) $f$ is analytic but not $g$

(b) $g$ is analytic but not $f$

(c) both are analytic

(d) neither $f$ nor $g$ is analytic

My attempt: for $f(z)$: $u_x=2x=2y=v_y$ and $u_y=0=-v_x$, hence $f$ is analytic at $0$, and for $g(z)$ we have: $u_x=2x=x=v_y$ and $u_y=2y=-y=-v_x$, hence $g$ is analytic at $z=0$.

But if we see the general definition, which implies that if $f$ is analytic at a point $z$ then it is analytic in all points of a neighborhood of $z$ then it follows that neither of them is analytic.

Is my approach correct and what should be the final solution to this? Thanks in advance.

Answer 1

CR equations have to hold in a neighborhood of $0$, as you have pointed out. The fact they are satisfied at a single point is meaningless.