Square root of a complex number with infinitesimal imaginary part I am going through lecture in Quantum Field Theory and I am encountering the square root of a complex number with infinitesimally small imaginary part: $$\lim_{y\rightarrow0^+}\sqrt{-x+iy}=-i\sqrt{x}$$ and $$\lim_{y\rightarrow0^+}\sqrt{-x-iy}=i\sqrt{x}$$
But I don't understand how these equalities are established.
The standard way of taking the square root is by writing the complex number in polar form and then the result is just the square root of the magnitude times the exponential of the imaginary unit times the argument of the complex number divided by two in addition to multiples of $\pi$. But when I take the limit of the imaginary part going to zero I just get a vanishing argument and I don't get the minus or plus sign in both the square roots. What am I not getting right here?

Ok I think I got it now thanks to peterwhy and Cyclotomic Field. Correct me if I am wrong.
$$\lim_{y\rightarrow 0+} \sqrt{-x\pm y}=\lim_{y\rightarrow 0+} \sqrt{x^2+y^2}e^{i\,arctan2(\pm y/-x)/2}$$
For the plus case: $$\lim_{y\rightarrow 0+} arctan2(y/-x)=\pi$$ and for the minus case: $$\lim_{y\rightarrow 0+} arctan2(-y/-x)=-\pi$$
Therefore $$\lim_{y\rightarrow 0+} \sqrt{-x\pm iy}=\mp i \sqrt{x}$$
