Here are a few functions for reference purposes:
$f(z) = \frac{sin(2z)}{z^3}$, $ \space g(z) = \frac{sin(z)}{tan(z)}$, $ \space h(z) = z^2 sin(\frac{1}{z})$
Suppose I was calculating the singularities of these functions at $z_0 = 0$ and I had to find the order of each function at $z_0$
I understand what all the different singularities are:- removable, pole and essential.
If I plot the graph of say $f(z)$ in the real plane where $z$ is a real number, it looks like the singularity is a pole. This is only a visual guess. It's definitely not removable.
It seems correctable ("fixable"), by multiplying $f$ by $\frac{z^3}{sin(2z)}$, but this would imply any bijective function with a singularity that is not removable must be a pole, right?
I don't think the function is essential, but my understanding of essential singularities is very weak.
When it comes to orders of a function at $z_0$, I know how to factorize polynomials and determine the order but I'm a bit stuck on the trigonometric functions.
Do I need to represent them as a polynomial using Taylor Series, or Laurent Series?
Also, is it a bad idea visualizing these complex functions as real functions.
If anyone has any better examples for explanation purposes, that's fine, I only have the functions above for referencing in the question.
Thanks in advance for any help.
My guesses for $g$ and $h$ are pole and essential respectively.