I am writting an article and i want to fix a choice for a square root of a complex number:
The square root that i want to choose satisfies
$(\sqrt{z})^2=z$ for all $z\in\mathbb{C}$ and $\sqrt{z}=\sqrt{z}$ if $z\in\mathbb{R}+$ and $\sqrt{z}=i\sqrt{-z}$ if $z\in\mathbb{R}-$
How can i explain that in the begining of the article ? I am not very good in english and i don't now how to formulate that once a time for all the article.
Please help me. Thanks in advance.
On
The conventions $\Re(\sqrt z)\gt0$ for $z\in\mathbb{R}^+$ and $\Im(\sqrt z)\gt0$ for $z\in\mathbb{R}^-$ are almost always adopted for a square root function. After that, there are two standard conventions:
$$\Re(\sqrt z)\gt0\quad\text{for }z\not\in\mathbb{R}$$ or $$\Im(\sqrt z)\gt0\quad\text{for }z\not\in\mathbb{R}$$
The first of these introduces a discontinuity in the complex-valued function $f(z)=\sqrt z$ at every point on the negative half of the real axis; the second does the same at every point on the positive half of the real axis.
The universally accepted convention (which isn't universally known) is that for positive real numbers the square root sign means the positive root. If you must, you can tell your readers that for negative real numbers $r$ you want $\sqrt{r}$ to be $i\sqrt{-r}$. You can just say that at the start.
For complex numbers with an imaginary part it's a bad idea to use the square root sign since there is no good convention for choosing one of the square roots over the other.
Related: Order of operations with complex numbers