# 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 of the complex variable given initially on a certain domain of the extended complex plane .

A pair consisting of a domain and a single-valued analytic, or holomorphic, function defined on 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 , which consists for of a power series

 (1)

and a disc of convergence, , with centre and radius . In the case , a Weierstrass element consists of a series

 (2)

and a domain of convergence of this series , .

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

It may happen that the initial element cannot be analytically continued to any point . In that case, is the natural domain of existence or domain of holomorphy of the function , while the boundary is the natural boundary of the function . For example, for the Weierstrass element

the natural boundary is the circle , the boundary of the disc of convergence , since this element cannot be analytically continued to any point such that . No matter what the domain , one can construct an analytic function for which is the natural domain of existence for and where the boundary is the natural boundary of (this follows, for example, from the Mittag-Leffler theorem).

The complete analytic function in its domain of existence 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 is a multi-valued function: For each point there exists, in general, an infinite set of elements with centre at this point. However, this set is at most countable (the theorem of Poincaré–Volterra). On the whole, the complete analytic function can be regarded as a single-valued analytic function only on the corresponding Riemann surface, which is a multi-sheeted covering surface over . For example, the complete analytic function is multi-valued in its domain of existence ; at each point it takes the countable set of values

and each point corresponds to a countable set of elements

with centre . Usually, one employs a single-valued branch of this complete analytic function, namely the principal value of the logarithm . This is a holomorphic function in the domain , and can be "continuously extended to -∞, 0" , i.e. for the limit

exists. (Likewise, exists; these limit values do not coincide (their difference is constant, equal to ).)

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:

 (3)

where is an integer and is a natural number, and the discs of convergence of these series are and . In particular, for and , 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 or . For and , the series of (3) define correspondingly unbranched and (algebraic) branched elements.

If under continuation of the initial element one allows for the occurrence of special elements, with series of the form (3), which in general are multi-valued (for ) and have singularities of pole-type (for ), one gets the Riemann domain of existence (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 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 and such that the domains and contain a common point are called equivalent at the point if there exists a neighbourhood around at which . This equivalence relation has the usual properties of reflexivity, symmetry and transitivity. An equivalence class of elements at a given point is called a germ of the analytic function at the point . The germ characterizes the local properties of the function at the given point. Two germs and are equal if any two representatives of the equivalence classes coincide in some neighbourhood of . Similarly, one can define arithmetic operations and differentiation on germs by means of representatives. The complete analytic function is the set of all germs , , of the analytic function obtained from a given by analytic continuation along all paths in . Equality of two complete analytic functions and and operations on complete analytic functions are defined as equality of the germs and at some point and operations on the germs.

The elements , the Weierstrass elements and the germs of analytic functions of several complex variables , , are defined exactly as above, but by means of domains in the complex space or by polydiscs of convergence,

of multiple power series

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