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.
$$\int_{+\partial D}\dfrac{\sin\left(\dfrac{1}{z}\right)\cos\left(\dfrac{1}{z-2}\right)}{z-5}\,\mathrm{dz}$$
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).
Then we get
$g(z)=\sum_{n=0}^{\infty}\frac{a_n}{(z-a)^n}$ for $z \ne a$ (Laurent).
Now you see: $a$ is an essential singularity of $g \iff f$ is not a polynomial.