Finding the Laurent Expansion of $1/z(1+z^2)$ when $1<\vert z \vert < \infty$

Question

Is the Laurent expansion in $1<\vert z \vert <\infty$ equivalent to the Laurent expansion at $\infty$? If so, $$f(1/z)=z/(1+1/z^2)=z^3/(1-(-z^2))=z^3-z^5+z^7-z^9+\cdots$$

However, wolframalpha says the expansion at infinity is $1/z^3-1/x^5-1/z^7+\cdots$

When I am asked to expand a Laurent series outside of some disc or annulus, should I just find the expansion at infinity?

Answer 1

You want an expansion of $f$ for $|z|>1$.

An easy way is to write $f(z) = {1 \over z^3} {1 \over 1+ {1 \over z^2}}$.

Since $|z|>1$, we have ${1 \over |z|^2 } < 1$ and so ${1 \over 1+ {1 \over z^2}} = 1 -{ 1\over z^2} + {1 \over z^4} - \cdots $.

Hence $f(z) = \sum_{n=0}^\infty (-1)^n {1 \over z^{n+3}}$.