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

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The discussion focuses on finding the Laurent series for the function cos(z)/z centered at z=0. The Maclaurin series for cos(z) is provided, which is the basis for deriving the Laurent series. The resulting Laurent series for cos(z)/z is expressed as a sum that includes a term 1/z and an infinite series. Participants are encouraged to ask further questions in a designated math help forum. This response aims to clarify the mathematical concept for the original poster.
Fernando Revilla
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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.
 
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