MHB Sammy's question at Yahoo Answers (Laurent expansion)

[Fernando Revilla]

Here is the question:

i have no idea...

Here is a link to the question:

Find Laurent series for cosz/z centered at z=0? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Sammy,

The Maclaurin expansion of $\cos z$ is: $$\cos z=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}\qquad (\forall z\in\mathbb{C})$$ so, the Laurent series expansion for $\cos z/z$ centered at $z=0$ is $$\frac{\cos z}{z}=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n-1}}{(2n)!}=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{(-1)^nx^{2n-1}}{(2n)!}\quad (0<|z|<+\infty)$$ If you have further questions, you can post them in the http://www.mathhelpboards.com/f50/ section.