Singularities in Functions

Question

Hi guys would someone mind dumbing down the terminology used for the types of singularities. I'm teaching myself complex analysis and can't seem to find any clear explanations of these. From what I'm seeing we're looking for points where the function is undefined examples I've come across:

$$\ i) f(z) = \frac 1 {z^4+z^2} $$ for this one I can see that when z = 0 it is undefined

$$\ ii) f(z) = e^{\frac {1}{(z+1)^2}} $$ for this I can see when z = -1

any help/ guidance would be greatly appreciated! Thanks

Answer 1

$i)$ Since $z^4+z^2=z^2(z^2+1)=z^2(z-i)(z+i)$, the function $f$ has poles in $0$,$i$ and $-i$. $f$ has in $0$ a pole of order $2$ and the poles $i$ and $-i$ are simple poles.

$ii)$ Take the Taylor expansion of $e^z$ and compute $e^{\frac {1}{(z+1)^2}}$. Then it is easy to see that $f$ has in $-1$ an essential singularity.

Answer 2

If a point is analytic, it can be assigned a Taylor series. However, when a point is not analytic, it cannot. It can instead be assigned a Laurent series. I suggest looking up how to find a Laurent series on Google.

The Laurent series tells you about the type of discontinuity of a particular function. I know the Taylor series for $e^x$, I can replace it with $x = \frac{1}{z²}$ to find it's Laurent series for z = 0. Then shift it by one to find your specific Laurent series. Because there are infinitely many terms that have negative powers of z, this is an infinite discontinuity.