I would like to show that $\operatorname{Arg}z$ is not an analytic function by using the Cauchy–Riemann differential equations.
After a quick search I've found this:
From Wikipedia: Alternatively, the principal value can be calculated in a uniform way using the tangent half-angle formula, the function being defined over the complex plane but excluding the origin:
$
\operatorname{Arg}(x + iy) =
\begin{cases}
2 \arctan \left( \frac{y}{\sqrt{x^2+y^2}+x} \right) & \qquad x > 0 \text{ or } y \ne 0 \\
\pi & \qquad x < 0 \text{ and } y = 0 \\
\text{undefined} & \qquad x = 0 \text{ and } y = 0
\end{cases}
$
Unfortunately, I was not able to make further progress. My question: Is it possible to show that $\operatorname{Arg}z$ is not an analytic function by using the Cauchy–Riemann differential equations?
Best Answer
As Neal points out in the comments, there's no need to use an explicit formula for $\arg$ because there is a more general, and very simple, principle at work here. Any complex-analytic function $\mathbb{C}\to\mathbb{R}$ must be constant, but the argument function is nonconstant. The Cauchy-Riemann equations tell us, because the imaginary part $v:=0$ is constant, that $u_x$ and $u_y$ are also $0$ and hence $u$ (the real part) is constant. This sidesteps any formal or technical work you would otherwise need to do here.