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**HallsofIvy** Write **what** as a Laurent series? How you do that depends strongly on the individual series. For an **analytic** function, of course, the Laurent series is just its Taylor series- there are no negative powers of x. A lot of the time, if you have some analytic function divided by powers of x, you can write the Laurent series for the whole function by starting with the Taylor series for the analytic "part" and then dividing each term by the the powers of x.

For example, if $\displaystyle f(x)= \frac{e^x}{x^2}$, start with the Taylor series $\displaystyle e^x= \sum_{n=0}^\infty \frac{1}{n!}x^n$ and divide each term by $\displaystyle x^2$: $\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^{n-2}$ which is the same (letting k= n-2) as $\displaystyle \sum_{k=-2}^\infty \frac{1}{(k+2)!}x^k$.

On the other hand, if you have other functions, like logarithm or tangent, that have worse singularities than just powers of x, the Laurent series is much, much harder to find.