Root of Complex Number 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.

Since $r\ge|x|$,
$$
\left(\sqrt{\dfrac{r+x}{2}}\right)^2=\dfrac{r+x}{2} \text{ and } \left(\sqrt{\dfrac{r-x}{2}}\right)^2=\dfrac{r-x}{2}
$$
However, since $y$ may be positive or negative,
$$
\sqrt{\dfrac{r+x}{2}}\sqrt{\dfrac{r-x}{2}}=\dfrac{\sqrt{r^2-x^2}}{2}=\dfrac{|y|}{2}
$$
Thus,
$$
\left(\sqrt{\dfrac{r+x}{2}}+i\sqrt{\dfrac{r-x}{2}}\right)^2=x+i|y|
$$
If $y>0$,
$$
\left(\sqrt{\dfrac{r+x}{2}}+i\sqrt{\dfrac{r-x}{2}}\right)^2=x+iy
$$
However, if $y<0$,
$$
\left(\sqrt{\dfrac{r+x}{2}}-i\sqrt{\dfrac{r-x}{2}}\right)^2=x+iy
$$
So to be correct, we should incorporate $\newcommand{\sgn}{\operatorname{sgn}}\sgn(y)$:
$$
\left(\sqrt{\dfrac{r+x}{2}}+i\sgn(y)\sqrt{\dfrac{r-x}{2}}\right)^2=x+iy
$$
Negate as necessary to get both solutions.
