# Holomorphic bijection between $\mathbb{C}-(-\infty,0]$ and $\{z\in\mathbb{C}:\mathrm{Re}(z)>0\}$

I'm looking for a holomorphic bijection between $$\mathbb{C}-(-\infty,0]$$ and $$\{z\in\mathbb{C}:\mathrm{Re}(z)>0\}$$. I know that $$\mathrm{Log}$$ (so the principal value) is a holomorphic bijection between $$\mathbb{C}-(-\infty,0]$$ and $$\{z\in\mathbb{C}:-\pi<\mathrm{Im}(z)<\pi\}$$. So then the required holomorphic bijection is just $$\exp(\frac12\mathrm{Log}(z))=\sqrt{z}$$, where this last expression is the principal value of the complex square root? Does this indeed suffice? Is the principal value of a complex square root always positive? The context: this question appears in my complex analysis syllabus but it seems odd to me that the solution is really this simple. Thanks in advance.

• A quick observation is that squaring the imaginary axis (boundary of the right half plane) produces $(-\infty,0]$ (boundary of the slit plane) – Hagen von Eitzen Feb 25 at 20:10
• @HagenvonEitzen okay thanks, and since the image of $\mathbb{C}-\{0\}$ under $z^2$ is $\mathbb{C}-\{0\}$, and $(-1)^2=1$, we see that the principal value of $\sqrt{z}$ indeed suffices? – Václav Mordvinov Feb 25 at 20:17