power series
power series
power series
pow′er se`ries
n.
| Noun | 1. | power series - the sum of terms containing successively higher integral powers of a variable |
n.
Power Series
power series
[′pau̇·ər ‚sir·ēz]Power Series
an infinite series of the form
a0 + a1z +a2z2 + . . . + anzn + . . .
where the coefficients a0, a1, a2, . . . , an, . . . are complex numbers independent of the complex variable z.
Generally speaking, the domain of convergence of a power series is an open region D = {z: ǀzǀ < R} bounded by a circle with center at z = 0. This circle is called the circle of convergence of the power series; its radius R is called the radius of convergence of the series. In the degenerate case where R = 0, the circle of convergence consists of the point z = 0. An example is 1 + l!z + 2!z2 + . . . + n!zn + . . . . If R = ∞, the domain of convergence is the entire complex plane. An example is 1 + z/1 ! + z2/2! + . . . + zn/n! + . . . . The radius of convergence of a power series is expressed in terms of its coefficients in accordance with the Cauchy-Hadamard theorem:
A power series converges absolutely at all points within the circle of convergence. On the circumference of the circle, where ǀzǀ = R, the series may either converge or diverge. For example, the series 1 + z + z2 + . . . + zn + . . . , for which R = 1, diverges at each point of the circumference, where ǀzǀ = 1. On the other hand, the series
converges absolutely at all points of the circumference, where ǀzǀ = 1. A power series diverges at each point exterior to the circle of convergence (ǀzǀ > R).
Within the circle of convergence, the sum of the power series
is an analytic function. Derivatives of any order of the function /(z) can be obtained by term-by-term differentiation of the series; moreover, the power series is the Taylor series of its sum.
A. A. GONCHAR
power series
- noun
Words related to power series
noun the sum of terms containing successively higher integral powers of a variable
Related Words
- series