Domain of Convergence

Domain of Convergence

 

the set of values of a variable x for which a series of functions

converges. The domain of convergence has a very simple form for power series. If a power series is considered for real values of the independent variable, then its domain of convergence is a single point, an interval (seeINTERVAL OF CONVERGENCE OF A POWER SERIES), which may contain one or both end points, or the entires-axis. If, however, complex values of the independent variable are also considered, then the domain of convergence of a power series is a single point, the interior of some circle (the circle of convergence), the interior of a circle and some points on the circumference, or the entire complex plane. Other types of series may have more complicated domains of convergence. For example, for a series of Legendre polynomials in the complex domain, the domain of convergence is the interior of an ellipse with foci at points — 1 and +1.

The domain of convergence is also defined for other processes. For example, the domain of convergence of an improper integral dependent on a parameter is understood to be the set of values of the parameter for which the given improper integral converges.

References in periodicals archive ?
The domain of convergence of the series is given by [absolute value of s] < [rho].
If 0 < [rho] < 1, then the domain of convergence is the interior of D.
The domain of convergence of the standard and modified asymptotic expansions can be analyzed by considering the singularities of the respective integrands in the complex plane, as shown in Section 2.