# Complex Variables. Complex square root function

## Homework Statement

Proof that $$f(z)=\sqrt{z}=e^{\frac{\ln z}{2}}$$ with logarithm branch $$[0,2\pi)$$. Then $$f$$ maps horizontal and vertical lines in $$A=\mathbb{C}-\{\mathbb{R}^{+}\cup\{0\}\}$$ on hyperbola branches.

## Homework Equations

I have that $$\ln_{[0,2\pi)} (z)=\ln\vert z\vert+i\mathop{\rm arg}\nolimits_{[0,2\pi)} (z)$$

## The Attempt at a Solution

I have tried to proof it directly, that means to describe vertical lines with $$z(y)=(a,y)$$ and horizontal ones with $$w(x)=(x,a)$$ and substitute in $$f$$. That produces an horrific expression that I can't reduce to an hyperbola. What can I do?