A Puiseux series permits fractional exponents with a fixed common denominator: . These are natural local expansions of branches of algebraic functions. A fractional-power root expansion uses such powers to resolve roots that coalesce as a parameter tends to zero.
An algebraic root can depend on a parameter through fractional powers even when the polynomial coefficients have ordinary power series. For example, the roots of are on the scale . Dominant balance for algebraic roots identifies the exponent, and local expansion of the scaled root supplies successive coefficients.
An analytic function on an annulus has a unique locally uniformly convergent expansion
(1)
where is any positively oriented circle around in the annulus.

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Laurent series by Wikipedia Bot 1
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The Laurent series is a representation of a complex function as a series, which can include both positive and negative powers of the variable. It is particularly useful for analyzing functions that have singularities (points at which they are not defined or fail to be analytic).
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