Complete analytic function

The set of all elements of an analytic function obtained by all analytic continuations (cf. Analytic continuation) of an initial analytic function $ f = f( z) $ of the complex variable $ z $ given initially on a certain domain $ D $ of the extended complex plane $ \overline{\mathbf C}\; $.

A pair $ ( D, f) $ consisting of a domain $ D \subset \overline{\mathbf C}\; $ and a single-valued analytic, or holomorphic, function $ f $ defined on $ D $ is called an element of an analytic function or an analytic element, or simply just an element. It is always possible when specifying an analytic function to use a Weierstrass or regular element $ ( U( a, R), f _ {a} ) $, which consists for $ a \neq \infty $ of a power series

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

and a disc of convergence, $ U( a, R) = \{ {z \in \overline{\mathbf C}\; } : {| z- a | < R } \} $, with centre $ a $ and radius $ R > 0 $. In the case $ a = \infty $, a Weierstrass element $ ( U( \infty , R), f _ \infty ) $ consists of a series

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

and a domain of convergence of this series $ U( \infty , R) = \{ {z \in \overline{\mathbf C}\; } : {| z | > R } \} $, $ R \geq 0 $.

Let $ E _ {f} $ be the set of all points $ \zeta \in \overline{\mathbf C}\; $ to which an initial element $ ( U( a, R), f _ {a} ) $ can be analytically continued over at least one path connecting the points $ a $ and $ \zeta $ in $ \overline{\mathbf C}\; $. One must bear in mind the possibility of the situation where for a point $ \zeta \in E _ {f} $ analytic continuation is possible along a certain class of paths $ L _ {1} $ but is impossible along any other class of paths $ L _ {2} $( see Singular point). The set $ E _ {f} $ is a domain in the plane $ \overline{\mathbf C}\; $. The complete analytic function (in the sense of Weierstrass) $ f _ {W} $ generated by the element $ ( U( a, R), f _ {a} ) $ is the name given to the set of all Weierstrass elements $ ( U( \zeta , R), f _ \zeta ) $, $ \zeta \in E _ {f} $, obtained by this kind of analytic continuation along all possible paths $ L \subset \overline{\mathbf C}\; $. The domain $ E _ {f} $ is called the (Weierstrass) domain of existence for the complete analytic function $ f _ {W} $. The use of an arbitrary element $ ( D, f) $ instead of a Weierstrass element leads to the same complete analytic function. The elements $ ( D, f) $ of $ f _ {W} $ are often called the branches of the analytic function $ f _ {W} $( cf. Branch of an analytic function). Any element $ ( D, f) $ of the complete analytic function $ f _ {W} $ taken as the initial one under analytic continuation leads to the same complete analytic function $ f _ {W} $. Each element $ ( U( \zeta , R), f _ \zeta ) $ of the complete analytic function $ f _ {W} $ can be obtained from any other element $ ( U( a, R), f _ {a} ) $ by analytic continuation along some path connecting the points $ a $ and $ \zeta $ in $ \overline{\mathbf C}\; $.

It may happen that the initial element $ ( D, f) $ cannot be analytically continued to any point $ \zeta \notin D $. In that case, $ D = E _ {f} $ is the natural domain of existence or domain of holomorphy of the function $ f $, while the boundary $ \Gamma = \partial D $ is the natural boundary of the function $ f $. For example, for the Weierstrass element

$$ \left ( U( 0, 1), f _ {0} ( z) = \sum _ { k= } 0 ^ \infty z ^ {k!} \right ) $$

the natural boundary is the circle $ \Gamma = \{ {z \in \overline{\mathbf C}\; } : {| z | = 1 } \} $, the boundary of the disc of convergence $ U( 0, 1) $, since this element cannot be analytically continued to any point $ \zeta $ such that $ | \zeta | \geq 1 $. No matter what the domain $ D \subset \overline{\mathbf C}\; $, one can construct an analytic function $ f _ {D} ( z) $ for which $ D $ is the natural domain of existence for $ f _ {D} ( z) $ and where the boundary $ \Gamma = \partial D $ is the natural boundary of $ f _ {D} ( z) $( this follows, for example, from the Mittag-Leffler theorem).

The complete analytic function $ f _ {W} $ in its domain of existence $ E _ {f} $ is, in general, not a function of points in the usual sense of the word. A situation frequently encountered in the theory of analytic functions is that the complete analytic function $ f _ {W} $ is a multi-valued function: For each point $ \zeta \in E _ {f} $ there exists, in general, an infinite set of elements $ ( U( \zeta , R), f _ \zeta ) $ with centre at this point. However, this set is at most countable (the theorem of Poincaré–Volterra). On the whole, the complete analytic function $ f _ {W} $ can be regarded as a single-valued analytic function only on the corresponding Riemann surface, which is a multi-sheeted covering surface over $ \overline{\mathbf C}\; $. For example, the complete analytic function $ f( z) = \mathop{\rm ln} z = \mathop{\rm ln} | z | + i \mathop{\rm arg} z $ is multi-valued in its domain of existence $ E _ {f} = \{ {z \in \overline{\mathbf C}\; } : {0 < | z | < \infty } \} $; at each point $ \zeta \in E _ {f} $ it takes the countable set of values

$$ f _ \zeta ( \zeta ; s) = \mathop{\rm ln} | \zeta | + i \mathop{\rm Arg} \zeta + 2 \pi si,\ \ s= 0, \pm 1 \dots $$

and each point $ \zeta \in E _ {f} $ corresponds to a countable set of elements

$$ ( U( \zeta , | \zeta | ), f _ \zeta ( z; s)) = $$

$$ = \ f _ \zeta ( \zeta ; s) + \sum _ { k= } 1 ^ \infty \frac{(- 1) ^ {k-} 1 }{k \zeta ^ {k} } ( z - \zeta ) ^ {k} $$

with centre $ \zeta $. Usually, one employs a single-valued branch of this complete analytic function, namely the principal value of the logarithm $ \mathop{\rm Ln} z = \mathop{\rm ln} | z | + i \mathop{\rm Arg} z $. This is a holomorphic function in the domain $ D = \{ {z \in \overline{\mathbf C}\; } : {0 < | z | < \infty, - \pi < \mathop{\rm arg} z < \pi } \} $, and can be "continuously extended to -∞, 0" , i.e. for $ \zeta \in ( - \infty , 0 ) $ the limit

$$ \lim\limits _ {\begin{array}{c} z \rightarrow \zeta \\ \mathop{\rm Im} z > 0 \end{array} } \ \mathop{\rm Ln} z = \mathop{\rm Ln} _ {+} z $$

exists. (Likewise, $ \lim\limits _ {z \rightarrow \zeta , \mathop{\rm Im} z < 0 } \mathop{\rm Ln} z = \mathop{\rm Ln} _ {-} z $ exists; these limit values do not coincide (their difference is constant, equal to $ 2 \pi i $).)

Inversion of the Weierstrass elements (1) and (2) (see Inversion of a series) gives rise to elements of more general nature, correspondingly defined by Puisieux series:

$$ \tag{3 } f _ {a} = \sum _ {k= \mu } ^ \infty c _ {k} ( z- a) ^ {k/ \nu } ,\ \ f _ \infty = \sum _ {k= \mu } ^ \infty c _ {k} z ^ {- k/ \nu } , $$

where $ \mu $ is an integer and $ \nu $ is a natural number, and the discs of convergence of these series are $ U( a, R) $ and $ U( \infty , R) $. In particular, for $ \mu \geq 0 $ and $ \nu = 1 $, the series (3) coincide with (1) and (2), which define regular elements; a difference from these is that the elements defined by the series (3) are called singular for $ \mu < 0 $ or $ \nu > 1 $. For $ \nu = 1 $ and $ \nu > 1 $, the series of (3) define correspondingly unbranched and (algebraic) branched elements.

If under continuation of the initial element $ ( U( a, R), f _ {a} ) $ one allows for the occurrence of special elements, with series of the form (3), which in general are multi-valued (for $ \nu > 1 $) and have singularities of pole-type (for $ \mu < 0 $), one gets the Riemann domain of existence $ E _ {R} $( which is larger than the Weierstrass domain of existence), and the correspondingly larger set of elements which are defined by series of the form (3) is called the analytic image. An analytic image differs from a complete analytic function by the addition of all singular elements obtained under extension of a given regular element. When a corresponding topology has been introduced, the analytic image becomes the Riemann surface for the given function.

For the construction of the complete analytic function $ f _ {W} $ one can use the concept of a germ of an analytic function instead of the concept of an element. It involves localizing the concept of an element, and discarding the radius of convergence, which in this case is not important. Two elements $ ( D, f) $ and $ ( G, h) $ such that the domains $ D $ and $ G $ contain a common point $ a $ are called equivalent at the point $ a $ if there exists a neighbourhood around $ a $ at which $ f \equiv h $. This equivalence relation has the usual properties of reflexivity, symmetry and transitivity. An equivalence class of elements at a given point $ a \in \overline{\mathbf C}\; $ is called a germ $ \mathbf f _ {a} $ of the analytic function at the point $ a $. The germ characterizes the local properties of the function at the given point. Two germs $ \mathbf f _ {a} $ and $ \mathbf g _ {a} $ are equal if any two representatives of the equivalence classes coincide in some neighbourhood of $ a $. Similarly, one can define arithmetic operations and differentiation on germs by means of representatives. The complete analytic function $ f _ {W} $ is the set of all germs $ \mathbf f _ \zeta $, $ \zeta \in E _ {f} $, of the analytic function obtained from a given $ \mathbf f _ {a} $ by analytic continuation along all paths in $ \overline{\mathbf C}\; $. Equality of two complete analytic functions $ f _ {W} $ and $ g _ {W} $ and operations on complete analytic functions are defined as equality of the germs $ \mathbf f _ {a} $ and $ \mathbf g _ {a} $ at some point $ a \in E _ {f} \cap E _ {g} $ and operations on the germs.

The elements $ ( D, f) $, the Weierstrass elements $ ( U ^ {n} ( a, R), f _ {a} ) $ and the germs of analytic functions of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n \geq 1 $, are defined exactly as above, but by means of domains $ D $ in the complex space $ \mathbf C ^ {n} $ or by polydiscs of convergence,

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

$$ R _ {j} > 0, j = 1 \dots n; \ a = ( a _ {1} \dots a _ {n} ); $$

$$ R = ( R _ {1} \dots R _ {n} ), $$

of multiple power series

$$ f _ {a} = f _ {a} ( z) = $$

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

The concept of a complete analytic function of several complex variables is constructed in complete analogy with the case of one variable.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[3]	G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) MR0122987 MR1530201 MR0092855 Zbl 0501.30039
[4]	B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian) MR0155003

Comments

The construct here called the analytic image is variously called the analytic configuration, the analytic entity and the analytische Gebilde in the English literature.

Additional references include H. Weyl's important monograph [a1] (translated into English as [a2]) and the more modern [a3].

References

[a1]	H. Weyl, "Die Idee der Riemannschen Fläche" , Teubner (1955) MR0069903 Zbl 0068.06001
[a2]	H. Weyl, "The concept of a Riemann surface" , Addison-Wesley (1955) (Translated from German) MR1440406 MR0166351
[a3]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 MR0583745 Zbl 0475.30001
[a4]	O. Foster, "Riemannsche Flächen" , Springer (1977)