Analytic function, element of an

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)

Comments

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

References

[a1]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4