I just want to clarify something here. Using elementary computation we can verify that for $x,y\in\mathbb{R}$
$\sqrt{x+iy}=\pm\left(\sqrt{\frac{r+x}{2}}+i \sqrt{\frac{r-x}{2}}\right)$
where $r=\sqrt{x^2+y^2}$.
However, in wikipedia the algebraic formula for the root is given by $\sqrt{x+iy}=\sqrt{\frac{r+x}{2}}\pm i \sqrt{\frac{r-x}{2}}$
By squaring $\sqrt{\frac{r+x}{2}}- i \sqrt{\frac{r-x}{2}}$ , We will get $x+iy=x-i\sqrt{y^2}$.
Wikipedia refers to the book Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Table , the book also giving the same formula as wikipedia's.
Is this something related to "Principal Value" or just a double typographic error?
I ask this because it is rare to see two same mistakes from two different sources, so I have a doubt.