Disc of convergence
of a power series
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The disc ,
, in which the series
is absolutely convergent, while outside the disc (for ) it is divergent. In other words, the disc of convergence
is the interior of the set of points of convergence of the series . Its radius
is called the radius of convergence of the series. The disc of convergence may shrink to the point
when
, and it may be the entire open plane, when
. The radius of convergence
is equal to the distance of the centre
to the set of singular points of
(for the determination of
in terms of the coefficients
of the series see Cauchy–Hadamard theorem). Any disc
,
, in the
-plane is the disc of convergence of some power series.
For a power series
![]() | (2) |
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in several complex variables ,
, a polydisc of convergence of the series (2) is defined to be any polydisc
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at all points of which the series (2) is absolutely convergent, while in any polydisc
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where and at least one of the latter inequalities is strict, there is at least one point at which the series is divergent. The radii
,
,
, of the polydisc of convergence are called the associated radii of convergence of the series (2). They are in a well-defined relationship with the coefficients of the series, so that any polydisc with centre
and with radii satisfying this relationship is the polydisc of convergence of a series (2) (cf. Cauchy–Hadamard theorem). Any polydisc
,
,
, in the complex space
is the polydisc of convergence for some power series in
complex variables. When
the interior of the set of points of absolute convergence of a series (2) is more complicated — it is a logarithmically convex complete Reinhardt domain with centre
in
(cf. Reinhardt domain).
References
[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) |
[2] | B.V. Shabat, "Introduction of complex analysis" , 1 , Moscow (1976) (In Russian) |
Comments
References
[a1] | L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 |
[a2] | L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1978) |
[a3] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1987) pp. 24 |
Disc of convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disc_of_convergence&oldid=17451