Singularities of complex functions.

Question

How do I determine the singularities of a function? What is a singularity? In the functions below which are the singularities?

a)$$f(z)=\frac{1}{(z^4+2z)}$$ b)$$f(z)={e^{1/z}}$$

Answer 1

Here is a start. For $(1)$ advance as

$$ f(z)=\frac{1}{(z^4+2z)}=\frac{1}{z(z^3+2)}. $$

$$z^3 = -2 = 2e^{-i\pi+i2k\pi} \implies z = 2^{1/3} e^{\frac{-i\pi+i2k\pi}{3}},\quad k=0,1,2. $$

Now you can see the singularities of $f(z)$. See here.

For $(2)$ check the notion of essential singularity.