Is this complex function differentiable at 0?

I'm moving my first steps in Complex Analysis. I can't tell whether the function
$$f(x+iy) = 2|xy|+i(y^2-x^2)$$
is differentiable at $0$ or not.

I tried using the limit of the difference quotient:
$$lim_zto 0frac+i(y^2-x^2)x+iy$$

The limit is $0$ when $xy>0$ because $x=-iy$ is a root of the numerator, but I can't compute the limit in general...

2 Answers
2

A less miraculous argument somewhat like the other answer: If $z=x+iy$ then it's clear that $|x|le|z|$ and $|y|le|z|$, hence $$|f(z)|le 2|z|^2+|z|^2+|z|^2.$$(So $$left|fracf(z)-f(0)zright|lefrac^2=4|z|to0quad(zto0).)$$

This seems preferable because for example it also applies to show the function $g(x+iy)=3|x||y|+x^2$ is differentiable at the origin, without the miraculous simplification.

Regarding the C-R equations, and some erroneous things that have been said in the comments: For this function we have $u_x=u_y=v_x=v_y=0$ at the origin, but the partials are not even defined in a neighborhood of the origin, so they are certainly not continuous at the origin, hence we cannot use the C-R equations to show $f$ is differentiable at the origin.

The theorem is this:

If $u_x,u_y,v_x$ and $v_y$ are continuous at $z$ then the C-R equations at $z$ are equivalent to differentiability at $z$.

Example showing that just knowing the C-R equations at a point does not imply differentiability at that point: Define $$f(x+iy)=begincases1,&(xy=0),
\0,&(xyne0).endcases$$

Then at the origin all four partials exist, and in fact $u_x=u_y=v_x=v_y=0$ at the origin, so the C-R equations are satisfied at the origin. But $f$ is not even continuous at the origin, so it's certainly not differentiable at the origin.

Ok, there is a "pointwise" version of the theorem that could be applied to the example in the question:

Suppose $f:Bbb CtoBbb C$ is Frechet differentiable at $z$ (when regarded as a map from $Bbb R^2$ to $Bbb R^2$) and satisfies the C-R equations at $z$. Then $f$ is complex differentiable at $z$.

The function in the question is Frechet differentiable at the origin. But this isn't really of any use, because verifying it's Frechet differentiable at $0$ is almost exactly the same as verifying directly that it's complex-differentiable at $0$.

HINT:

$$left|fracyx+iyright|=sqrtfrac4x^2y^2+(y^2-x^2)^2x^2+y^2=sqrtx^2+y^2$$

Can you prove that the limit exists?

Required, but never shown

Required, but never shown

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