How do singularities of a function on a complex plane affect real line behavior?

[Simfish]

consider the function

$$\frac{1}{\epsilon^2 + z^2}$$

So we know that there are two poles, one at $$z = i \epsilon$$, one at $$z = - i \epsilon$$. So when this function never hits 0 on the real line, how do the singularities affect its behavior on the line?

Okay, so poles are a subclass of singularities. I think that $$z = i \epsilon$$ and $$z = - i \epsilon$$ are poles - I may be wrong here. The question is - how do complex singularities (complex poles in this case) affect a function's behavior when the function is plotted on the real line?

 

[mathwonk]

they determine the radius of convergence of the real taylor series for the function, in this case it is e, about x=0.