# Laurent Series for singularities and poles

Hi guys I was wondering how I can understand if the sin and the cos has essential singularities. for instance if I want to understand if 0 which singularity is i, can write the Laurent series only of the sin (centred in 0) and see how it works , or MUST write the Laurent series of all the function (centered in zero) ? Same for cos , help I want to understand this topic very well. Thk.

$$\int_{+\partial D}\dfrac{\sin\left(\dfrac{1}{z}\right)\cos\left(\dfrac{1}{z-2}\right)}{z-5}\,\mathrm{dz}$$

• ok sorry , i was tryng to write the laurent series but it's difficoult so i want to understand if i have to write the resultin series of all the functions o i can just write the series of the sin( for instance) to understan which singularity is – xmaionx Jul 20 '18 at 9:04
• you only need to write down the Laurent series for $\sin (1/z)$ centered at $0$. The other part $\cos (1/(z-2))/(z-5)=c_0+ c_1z+c_2z^2+\text{...}$ is analytic inside the unit disk. Then multiply the two series – Lozenges Jul 20 '18 at 10:38
• @Lozenges so for instance if i want to see if z=2 is an essential singularity i just wtrite the cos (1/(z-2)) series centered in 2 ?? – xmaionx Jul 24 '18 at 14:56

Suppose that $f$ is an entire function, let $a \in \mathbb C$ and define $g(z):=f(\frac{1}{z-a})$ for $z \ne a$.
We have $f(z)= \sum_{n=0}^{\infty}a_nz^n$ for all $z$ (Taylor).
$g(z)=\sum_{n=0}^{\infty}\frac{a_n}{(z-a)^n}$ for $z \ne a$ (Laurent).
Now you see: $a$ is an essential singularity of $g \iff f$ is not a polynomial.