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

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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 Analysis section.