Disc of convergence

of a power series

$$ \tag{1 } f ( z) = \ \sum _ {k = 0 } ^ \infty c _ {k} ( z - a) ^ {k} $$

The disc $ \Delta = \{ {z } : {| z - a | < R } \} $, $ z \in \mathbf C $, in which the series

is absolutely convergent, while outside the disc (for $ | z - a | > R $) it is divergent. In other words, the disc of convergence $ \Delta $ is the interior of the set of points of convergence of the series . Its radius $ R $ is called the radius of convergence of the series. The disc of convergence may shrink to the point $ a $ when $ R = 0 $, and it may be the entire open plane, when $ R = \infty $. The radius of convergence $ R $ is equal to the distance of the centre $ a $ to the set of singular points of $ f ( z) $( for the determination of $ R $ in terms of the coefficients $ c _ {k} $ of the series see Cauchy–Hadamard theorem). Any disc $ \Delta = \{ {z } : {| z | < R } \} $, $ 0 \leq R \leq \infty $, in the $ z $- plane is the disc of convergence of some power series.

For a power series

$$ \tag{2 } f ( z) = \ f ( z _ {1} \dots z _ {n} ) = $$

$$ = \ \sum _ {k _ {1} \dots k _ {n} = 0 } ^ \infty c _ {k _ {1} \dots k _ {n} } ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} } $$

in several complex variables $ z _ {1} \dots z _ {n} $, $ n > 1 $, a polydisc of convergence of the series (2) is defined to be any polydisc

$$ \Delta _ {n} = \ \{ {z = ( z _ {1} \dots z _ {n} ) } : { | z _ \nu - a _ \nu | < R _ \nu ,\ \nu = 1 \dots n } \} $$

at all points of which the series (2) is absolutely convergent, while in any polydisc

$$ \{ {z = ( z _ {1} \dots z _ {n} ) } : { | z _ \nu - a _ \nu | < R _ \nu ^ { \prime } ,\ \nu = 1 \dots n } \} , $$

where $ R _ \nu ^ { \prime } \geq R _ \nu $ and at least one of the latter inequalities is strict, there is at least one point at which the series is divergent. The radii $ R _ \nu $, $ \nu = 1 \dots n $, $ 0 \leq R _ \nu \leq \infty $, 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 $ a $ and with radii satisfying this relationship is the polydisc of convergence of a series (2) (cf. Cauchy–Hadamard theorem). Any polydisc $ \Delta _ {n} $, $ 0 \leq R _ \nu \leq \infty $, $ \nu = 1 \dots n $, in the complex space $ \mathbf C ^ {n} $ is the polydisc of convergence for some power series in $ n $ complex variables. When $ n > 1 $ 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 $ a $ in $ \mathbf C ^ {n} $( 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)

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