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Puiseux Series

A Puiseux series is a formal power series or Laurent series in a fixed fractional power of its variable. About z=z_0, it has the form

sum_(k=k_0)^inftya_k(z-z_0)^(k/n),

where n is a positive integer and k_0 is an integer (Duval 1989, p. 122). The exponents have a common denominator, and only finitely many negative powers occur. The substitution z-z_0=t^n converts it to a Laurent series in t. Logarithms and nested logarithms are not part of the classical definition, although they occur in more general series expansions.

Puiseux's theorem ensures that the local branches of an algebraic function over C admit convergent Puiseux expansions. For example, the equation w^2=z^3(1+z) has the two expansions

w_+/-(z)=+/-z^(3/2)(1+z/2-(z^2)/8+(z^3)/(16)+...),

which converge for |z|<1 on a chosen branch of z^(1/2). Continuing once around the origin interchanges them. The smallest denominator n is the ramification index of the branch.

See also

Algebraic Branch Point, Algebraic Function, Formal Power Series, Laurent Series, Maclaurin Series, Power Series, Puiseux's Theorem, Ramification Index, Taylor Series

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References

Davenport, J. H.; Siret, Y.; and Tournier, E. Computer Algebra: Systems and Algorithms for Algebraic Computation, 2nd ed. San Diego, CA: Academic Press, pp. 90-92, 1993.Duval, D. "Rational Puiseux Expansions." Compositio Math. 70, 119-154, 1989. https://www.numdam.org/item/CM_1989__70_2_119_0/.Koepf, W. "Closed Form Laurent-Puiseux Series of Algebraic Functions." Appl. Algebra Eng. Commun. Comput. 7, 21-26, 1996. https://doi.org/10.1007/BF01613613.McDonald, J. "Fiber Polytopes and Fractional Power Series." J. Pure Appl. Algebra 104, 213-233, 1995. https://doi.org/10.1016/0022-4049(94)00129-5.McDonald, J. "Fractional Power Series Solutions for Systems of Equations." Disc. Comput. Geom. 27, 501-529, 2002. https://doi.org/10.1007/s00454-001-0077-0.McMullen, C. T. "Riemann Surfaces." Harvard University, Math 213b, Apr. 28, 2026. https://abel.math.harvard.edu/~ctm/math213b/home/course/course.pdf.Puiseux, V. "Recherches sur les fonctions algébriques." J. Math. Pures Appl. 15, 365-480, 1850. https://www.numdam.org/item/JMPA_1850_1_15__365_0/.Siegel, C. L. Topics in Complex Function Theory, Vol. 1: Elliptic Functions and Uniformization Theory. New York: Wiley, p. 98, 1988.Trott, M. "Series Expansions." §1.6.4 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Puiseux Series

Cite this as:

Weisstein, Eric W. "Puiseux Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PuiseuxSeries.html

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