Laurent Series of $\sin((z+2)/z)$

Question

i'm having trouble with this question, im not sure if I can develop more.

Determine the Laurent series centered in $0$ of the function $f(z)=sin((z+2)/z)$ with U= {$0$ < |z| < ∞}

So we have: $$\sin((z+2)/z)=\sin(1+2/z)=\sin(1)\cos(2/z)+\cos(1)\sin(2/z)$$

From now, I can develop $\cos(2/z)$ and $\sin(2/z)$ into series and remplace them but that's not how a Laurent serie should look like. Thanks for your suggestions!!

Answer 1

Yes, you can develop $\cos\left(\frac2z\right)$ and $\sin\left(\frac2z\right)$ into series, and, in fact, that's what you should do. You will get that\begin{align}\sin\left(\frac{z+2}z\right)&=\sin(1)\sum_{n=0}^\infty\frac{(-1)^n2^{2n}}{(2n)!z^{2n}}+\cos(1)\sum_{n=0}^\infty\frac{(-1)^n2^{2n+1}}{(2n+1)!z^{2n+1}}\\&=\sin(1)+\frac{2\cos(1)}z-\frac{2\sin(1)}{z^2}-\frac{4\cos(1)}{3z^3}+\cdots\end{align}