I have a doubt with the singularities and poles of a function, for example in the next complex function:

$\displaystyle f(x)=\sqrt{x^2+a}$

with $\displaystyle a$ a constant. The function is zero when $\displaystyle x=\pm{\sqrt{a}}$

For this value of x, is it a singularity or a pole? If it is a pole, of which order? Is it possible to calculate the residue for $\displaystyle x=\pm{\sqrt{a}}$?