Analytic function, element of an

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The collection $( D , f )$ of domains $D$ in the plane $\mathbf C$ of a complex variable $z$ and analytic functions $f (z)$ given on $D$ by a certain analytic apparatus that allows one to effectively realize the analytic continuation of $f (z)$ to its whole domain of existence as a complete analytic function. The simplest and most frequently used form of an element of an analytic function is the circular element in the form of a power series

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

and its disc of convergence $D = \{ {z \in \mathbf C } : {| z - a | < R } \}$ with centre at $a$( the centre of the element) and radius of convergence $R > 0$. The analytic continuation here is achieved by a (possibly repeated) re-expansion of the series (1) for various centres $b$, $| b - a | \leq R$, by formulas like

$$f (z) = \sum _ { n=0 } ^ \infty d _ {n} ( z - b ) ^ {n} = \ c _ {0} +$$

$$+ [c _ {1} ( b - a ) + c _ {1} ( z - b ) ] +$$

$$+ [ c _ {2} ( b - a ) ^ {2} + 2 c _ {2} ( b - a ) ( z - b ) + c _ {2} ( z - b ) ^ {2} ] + \dots .$$

Any one of the elements $( D , f )$ of a complete analytic function determines it uniquely and can be represented by means of circular elements with centres $a \in D$. In the case of the centre at infinity, $a = \infty$, the circular element takes the form

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

with domain of convergence $D = \{ {z \in \mathbf C } : {| z | > R } \}$.

In the process of the analytic continuation, $f (z)$ may turn out to be multiple-valued and there may appear corresponding algebraic branch points (cf. Algebraic branch point), that is, branched elements of the form

$$f (z) = \sum _ { k=m } ^ \infty c _ {k} ( z - a ) ^ {k / \nu } ,$$

$$f (z) = \sum _ { k=m } ^ \infty c _ {k} z ^ {- k / \nu } ,$$

where $\nu > 1$; the number $\nu - 1$ is called the branching order. The branched elements generalize the concept of an element of an analytic function, which in this connection is also called an unramified (for $\nu = 1$) regular (for $m \geq 0$) element.

As the simplest element $( D , f )$ of an analytic function $f (z)$ of several complex variables $z = ( z _ {1} \dots z _ {n} )$, $n > 1$, one can take a multiple power series

$$\tag{2 } f (z) = \sum _ {| k | = 0 } ^ \infty c _ {k} ( z - a ) ^ {k\ } =$$

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

where $a = ( a _ {1} \dots a _ {n} )$ is the centre, $| k | = k _ {1} + \dots + k _ {n}$, $c _ {k} = c _ {k _ {1} } \dots c _ {k _ {n} }$, $( z - a ) ^ {k} = ( z - a _ {1} ) ^ {k _ {1} } \dots ( z - a _ {n} ) ^ {k _ {n} }$, and $D$ is some polydisc

$$D = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} - a _ {j} | < R _ {j} ,\ j = 1 \dots n } \}$$

in which the series (2) converges absolutely. However, for $n > 1$ one has to bear in mind that a polydisc is not the exact domain of absolute convergence of a power series.

The concept of an element of an analytic function is close to that of the germ of an analytic function.

References

 [1] A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian) [2] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

For $n > 1$ the domain of absolute convergence of a power series is a so-called Reinhardt domain, cf. [a1].