power series

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power series

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Alternative forms

• power-series (attributive use)

Noun

power series (plural power series)

1. (mathematics, mathematical analysis) Any infinite series of the general form ${\displaystyle \sum _{i=0}^{\infty }a_{i}(x-c)^{i}}$.
   • 1983, Ian Stewart, David Tall, Complex Analysis, Cambridge University Press, page 259:

     Thus no single choice of z₀ will give a power series expansion of f(z) valid for all ${\displaystyle \textstyle z\in \mathbb {C} \setminus \left\{-1,1\right\}}$ even though f is analytic on this set.

   • 1988, Richard Courant, translated by E. J. McShane, Differential and Integral Calculus, 2nd edition, volume 1, Wiley, page 413:

     In this we may use as our starting-point a general discussion of the theory of power series with complex variables and complex coefficients. The construction of such a theory of power series offers no difficulty once we define the concept of limit in the domain of complex numbers; in fact, it follows the theory of real power series almost exactly.

   • 1899, Oskar Bolza, The Theory of Functions, 2013, Reprint, Books on Demand, page 69,

     From this theorem (for which in many cases Cauchy's theorem on double sums may be substituted) follow easily the rules for the multiplication and division of power series, Taylor's theorem for power series along with the rules for differentiation of power series and series of power series.

Usage notes

The constant c is sometimes called the centre of the power series, and is often 0.

Hyponyms

• Maclaurin series, Taylor series
• Laurent series

Derived terms

• formal power series

Translations

References

• Oregon State University Mathematics Dept Web study guide - Introduction to Power Series
• Power Series, Folkmar Bornemann, Eric W. Weisstein, MathWorld