Classifying singularities and determing orders of complex functions

Question

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.

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My guesses for $g$ and $h$ are pole and essential respectively.

Answer 1

A singularity of a holomorphic function $w$ at a point $a$ is:

• Removable if the limit of $w$ at $a$ exists, and if $w$ were extended to have the limit as its value at $a$, then the extended function is holomorphic at $a$. ("Holomorphic at $a$" = "analytic at $a$" = "derivative exists in some neighborhood of $a$".)
• A pole, if it is not removable, but there is a positive integer $n$ such that $(z - a)^nw(z)$ has a removable singularity at $a$. The order of the pole (not the function) is the smallest value of $n$ that works.
• Essential if it is not removable nor a pole.

I.e.,

• A removable singularity is one that doesn't really need to be there. It is just an artifact of how the function is defined, not a natural part of the function's behavior.
• A pole is a real singularity, but one that is well-behaved. In some neighborhood of the singularity, the function acts like $\frac1{z^n}$ does near $0$.
• An essential singularity is not well-behaved. Unlike poles, it may not be isolated, and even when it is isolated, its behavior is complicated (per Picard's theorem, it is very complicated).

For your functions,

• Note that multiplying $f(z)$ by $z^3$ gives a function with an obviously removable singularity, so $f(z)$ has a pole. The order of that pole is at most $3$, but in fact it is smaller (I'll let you try to figure out what it really is).
• For $g(z)$, recall that $\tan z = \frac {\sin z}{\cos z}$ and simplify. How does $\cos z$ behave at $0$? (Note that $g(z)$ itself is not defined at $0$, so it does have a singularity there.)
• While $\lim_{z\to0} h(z)$ exists, consider $\lim_{z\to0} h'(z)$. If it is analytic, all derivatives of $h$ would exist. Multiplying $h(z)$ by $z^n$ just puts off the derivative where the limit fails to exist. It does not solve the issue altogether.

Your concept on $f$ that it is fixable by multiplying by $\frac {z^3}{\sin 2z}$ is of no value. Every singularity of any analytic function is "fixable" by multiplying by the function's inverse! This is not a useful property.