# Use the Cauchy Riemann Equations to determine whether a function is analytic (Complex Analysis)

Is the function $$f(z)=|z^{2}-z|$$ nowhere analytic? Justify your ansewr

WHat i tried

Let $z=x+iy$ and then substituting it to the function above to get the form $$f(x+iy)=u(x,y)+i(x,y)$$ where $u(x,y)$ is the real part and $v(x,y)$ the

imaginary part and then using the Cauchy Riemann equations to show that it holds.ie

$$U_{x}=V_{y}$$ $$U_{y}=-V_{x}$$ Then find the four partial derivatives $U_{x}$, $U_{y}$ , $V_{x}$, $V_{y}$ and if they are continuous then the function are analytic. My problem is that because of the modulus i couldent differentiate and find the partial derivatives in the usual way. Do i have to split up the modulus into the nehgative and positive portion and find the partial derivatives for each case. Could anyone explain. Thanks

• yes, do a case distinction – noctusraid Sep 5 '16 at 11:18
• "nowhere analytic" means "no open set where the function is complex differentiable"? – zhw. Sep 5 '16 at 19:06

Hint: The modulus of a complex number $a+ib$ is $\sqrt{a^2+b^2}$. Use this to identify $u$ and $v$ as functions of $x,y$. Hint2: There is one point at which $f$ is complex differentiable.
• Indeed, so what is $v(x,y)$ ? – H. H. Rugh Sep 5 '16 at 11:50
• Precisely! So now you go on calculating derivatives of $u$ and compare... – H. H. Rugh Sep 5 '16 at 12:13