Apply Cauchy-Riemann equation on $\sqrt{z}$ to show it is analytic

Question

I want to show that the function $f(z) = \sqrt{z}$ is analytic on $D = \{z\in\mathbb{C}:Re(z)>0\}$ by the Cauchy-Riemann equation.

But here is the thing. I fail to rewrite $f(z)$ into $u(x,y) + iv(x,y)$. So can anyone help me to rewrite it?

Is $\sqrt{z}$ an analytic function? Here I found that $f(z)$ is analytic, but they did not use Cauchy-Riemann which I want to use.

Thanks in advance.

Answer 1

I think $\sqrt{(x+\sqrt{x^2+y^2})/2}+i\,\text{sgn}(y)\sqrt{(-x+\sqrt{x^2+y^2})/2}$ is the expression for the 'positive' root of $x+iy$ that you are looking for. See e.g. here for more info. I'm not sure that it'll be economical to prove the holomorphicity of $\sqrt{z}$ by verifying the Cauchy-Riemann equations starting from this expression.