What is the nature of the singularity of this function?

Question

If I've been asked to determine the nature of the singularity of the function $\frac{1}{z^2+a^2}$, where $z$ and $a$ have not been defined but from the context I assume $z$ is a complex variable and $a$ is a constant, then I think it will be undefined at $y^2 = x^2+2xiy+a^2$. I can't make that into the form $y=f(x)$, so I don't know what to say about the nature of the singularity, have I incorrectly identified the point where it occurs?

Answer 1

The singularity occurs at two points: $z=\pm ai$, since these are the values of $z$ such that $z^2+a^2=0$. Since$$\frac1{z^2+a^2}=\frac1{(z+ai)(z-ai)}$$both singularities are simple poles.