Basic question on Determing Singular Points

[RJLiberator]

Determine the singular points of each function:

f(z) = (z^3+i)/(z^2-3z+2)

So it is my understanding that a singular point is one that makes the denominator 0 in this case.
We see that (z-2)(z-1) is the denominator and we thus conclude that z =2, z=1 are singular points.

f(z) = (2z+1)/(z(z^2+1))

So, z=0, +/- i are singular points.

Am I understanding this correctly?

 

[RUber]

That is how I understand it. In general if you can find a value of z for which f(z) is undefined that is a singularity. You should also check what happens in the numerator to be sure you aren't missing anything.

 

[RUber]

The formal definition for isolated singularities (or singular points), as stated in Fischer's "Complex Variables" is:
"An analytic function f has an isolated singularity at a point z_0 if f is analytic in the punctured disc 0<|z-z_0|<r, for some r>0.
That is, the function is well-defined in the neighborhood of the point, but not at the point itself.

 

[HallsofIvy]

Given that f(z) is a rational function, a polynomial divided by a polynomial, then all singular points are where the denominator is 0.

 

[RJLiberator]

Excellent. Thank you for this confirmation.

This class is moving fasttttt.

=)