Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and This article is about both real and complex analytic functions Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with

The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology. The theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 In Mathematics, sheaf cohomology is the aspect of Sheaf theory, concerned with sheaves of Abelian groups that applies Homological algebra to

1 Initial discussion
2 Applications
3 Formal definition of a germ
4 The topology of the set of germs
5 Examples of analytic continuation
6 Monodromy theorem
7 Hadamard's gap theorem
8 Polya's theorem
9 See also

Initial discussion

Analytic continuation of natural logarithm (imaginary part)

Suppose f is an analytic function defined on an open subset U of the complex plane C. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis If V is a larger open subset of C, containing U, and F is an analytic function defined on V such that

F(z) = f(z) for all z in U,

then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.

Analytic continuations are unique in the following sense: if V is connected and is the domain of both F₁ and F₂, two analytic continuations of f, then

F₁ = F₂

everywhere. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" That is because the difference is an analytic function which vanishes on a non-empty open set U (the domain of f), and an analytic function which vanishes on a non-empty open set must vanish everywhere on its domain (assuming the domain is connected) and hence must be identically zero. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of

For example, if a power series with radius of convergence r about a point a of C is given, one can consider analytic continuations of the power series, i. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a e. analytic functions F which are defined on larger sets than the open disc of radius r at a, in symbols

{z : |z − a| < r},

and agree with the given power series on that set. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. The number r is maximal in the following sense: there always exists a complex number z with

|z − a| = r

such that no analytic continuation of the series can be defined at z. Therefore there is a limitation to analytic continuation to bigger discs with the same centre a. On the other hand there may well be analytic continuations to some larger sets. That depends on the radius of convergence when you expand about points b other than a; if that is greater than

r − |b − a|

then we win the right to use that expansion on an open disc, part of which lies outside the original disc of definition. If not, there is a natural boundary on the bounding circle.

Applications

A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, and then extending it by analytic continuation. In practice, this continuation is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values Examples are the Riemann zeta function and the gamma function. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function

The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism This article is about both real and complex analytic functions The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional

The power series defined above is generalized by the idea of a germ. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable The general theory of analytic continuation and its generalizations are known as sheaf theory. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

Formal definition of a germ

Let

$f(z)=\sum_{k=0}^\infty \alpha_k (z-z_0)^k$

be a power series converging in the disc D_r(z₀) := {z in C : |z - z₀| < r} for r > 0. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + (Note, without loss of generality, here and in the sequel, we will always assume that a maximal such r was chosen, even if it is ∞. ) Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector

g = (z₀, α₀, α₁, α₂, . . . )

is a germ of f. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable The base g₀ of g is z₀, the stem of g is (α₀, α₁, α₂, . . . ) and the top g₁ of g is α₀. The top of g is the value of f at z₀, the bottom of g.

Any vector g = (z₀, α₀, α₁, . . . ) is a germ if it represents a power series of an analytic function around z₀ with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs $\mathcal G$ .

The topology of the set of germs

Let g and h be germs. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable If |h₀ - g₀| < r where r is the radius of convergence of g and if the power series that g and h represent define identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write g ≥ h. This compatibility condition is neither transitive, symmetric nor antisymmetric. If we extend the relation by transitivity, we obtain a symmetric relation, which is therefore also an equivalence relation on germs (but not an ordering). In Mathematics, the transitive closure of a Binary relation R on a set X is the smallest Transitive relation on X In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted $\cong$ .

We can define a topology on $\mathcal G$ . Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Let r > 0, and let

$U_r(g) = \{h \in \mathcal G : g \ge h, |g_0 - h_0| < r\}.$

The sets U_r(g), for all r > 0 and g ∈ $\mathcal G$ define a basis of open sets for the topology on $\mathcal G$ . In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets

A connected component of $\mathcal G$ (i. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of e. , an equivalence class) is called a sheaf. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. We also note that the map φ_g(h) = h₀ from U_r(g) to C where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas for $\mathcal G$ , hence $\mathcal G$ is a Riemann surface. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional $\mathcal G$ is sometimes called the universal analytic function.

Examples of analytic continuation

$L(z) = \sum_{k=1}^\infin \frac{(-1)^{k+1}}{k}(z-1)^k$

is a power series corresponding to the natural logarithm near z = 1. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational This power series can be turned into a germ

$g=(1,0,1,-\frac 1 2, \frac 1 3 , - \frac 1 4 , \frac 1 5 , - \frac 1 6 , \cdots)$

This germ has a radius of convergence of 1, and so there is a sheaf S corresponding to it. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. This is the sheaf of the logarithm function.

The uniqueness theorem for analytic functions also extends to sheaves of analytic functions: if the sheaf of an analytic function contains the zero germ (i. e. , the sheaf is uniformly zero in some neighborhood) then the entire sheaf is zero. Armed with this result, we can see that if we take any germ g of the sheaf S of the logarithm function, as described above, and turn it into a power series f(z) then this function will have the property that exp(f(z))=z. If we had decided to use a version of the inverse function theorem for analytic functions, we could construct a wide variety of inverses for the exponential map, but we would discover that they are all represented by some germ in S. In that sense, S is the "one true inverse" of the exponential map.

In older literature, sheaves of analytic functions were called multi-valued functions. In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map See sheaf for the general concept. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

Monodromy theorem

Main article: Monodromy theorem

The monodromy theorem gives a sufficient condition for the existence of a direct analytic continuation (i. In Mathematics, more precisely in Complex analysis, the monodromy theorem is an important result about Analytic continuation of a complex-analytic e. , an extension of an analytic function to an analytic function on a bigger set).

Suppose D is an open set in $\mathbb{C}$ , and f an analytic function on D. If G is a simply connected domain containing D, such that f has an analytic continuation along every path in G, starting from some fixed point a in D, then f has a direct analytic continuation to G. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be

In the above language this means that if G is a simply connected domain, and S is a sheaf whose set of base points contains G, then there exists an analytic function f on G whose germs belong to S.

Hadamard's gap theorem

Main article: Ostrowski-Hadamard gap theorem‎

For a power series

$f(z)=\sum_{k=0}^\infty \alpha_k (z-z_0)^k$

with coefficients mostly zero in the precise sense that they vanish outside a sequence of exponents k(i) with

$\lim_{i\to\infty} \frac{k(i+1)}{k(i)} > 1 + \delta \,$

for some fixed δ > 0, the circle centre z₀ and with radius the radius of convergence is a natural boundary. Such a power series defines a lacunary function. In analysis, a lacunary function, also known as a lacunary series, is an Analytic function that cannot be analytically continued anywhere outside

Polya's theorem

Let $f(z)=\sum_{k=0}^\infty \alpha_k (z-z_0)^k$ be a power series, then there exist $\epsilon_k\in \{-1,1\}$ such that

$f(z)=\sum_{k=0}^\infty \epsilon_k\alpha_k (z-z_0)^k$

has the convergence disc of f around z₀ as a natural boundary.

The proof of this theorem makes use of Hadamard's gap theorem.

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