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.

 

[tacman]

Firstly, it is important to note that the function \frac{1}{\epsilon^2 + z^2} is undefined at the points z = i \epsilon and z = - i \epsilon, since the denominator becomes zero at these points. These points are known as singularities of the function.

On the real line, the function is defined for all real values of z except for z = \pm \epsilon. This means that the function has a discontinuity at these points, as it is not defined at these points. The behavior of the function on the real line is affected by these singularities in the following ways:

1. Discontinuity: As mentioned before, the function has a discontinuity at z = \pm \epsilon, which means that the function is not continuous at these points. This can be seen when plotting the function on the real line, as there will be a gap at these points.

2. Asymptotic behavior: When approaching the singularities from the real line, the function approaches infinity. This is because as z gets closer to \pm \epsilon, the denominator becomes smaller and smaller, causing the overall value of the function to increase towards infinity.

3. Influence on nearby points: The singularities can also affect the behavior of the function at nearby points on the real line. This is because the function is undefined at the singularities, and any point close to the singularities will also have undefined values. This can cause unexpected behavior of the function in this region.

In conclusion, the singularities of a function on a complex plane can have a significant impact on the behavior of the function when plotted on the real line. They can cause discontinuities, affect the asymptotic behavior, and influence the behavior of nearby points. It is important to consider these singularities when analyzing the behavior of a function on the real line.