# Disc of convergence

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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)