Analytic Continuation

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 . 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 . INITIAL DISCUSSION Suppose ''f'' is an analytic function defined on an Open Subset ''U'' of the Complex Plane C. If ''V'' is a larger open subset of C, containing ''U'', and ''F'' is an analytic function defined on ''V'' such that F 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 ''F''₁ and ''F''₂, two analytic continuations of ''f'', then F everywhere. 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. 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.e. analytic functions ''F'' which are defined on larger sets than the Open Disc of radius ''r'' at ''a'', in symbols
	:''z'' &minus ''a''	''r''
	Be A	"http://wwwinformationdelightinfo/information/entry/power_series" class="copylinks">Power Series converging in the disc ''D''<sub>''r''</sub>(''z''<sub>0</sub>) := {''z'' in '''C''' : ''z'' - ''z''<sub>0</sub> < ''r''} for ''r'' > 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
	Let ''g'' And ''h'' Be	"http://wwwinformationdelightinfo/information/entry/germ_(mathematics)" class="copylinks">Germs If ''h''<sub>0</sub> - ''g''<sub>0</sub> < ''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) This extension by transitivity is one definition of analytic continuation The equivalence relation will be denoted <math>\cong</math>
	:<math>U R(g)	\{h \in \mathcal G : g \ge h, g_0 - h_0 < r\}</math>

is a power series corresponding to the Natural Logarithm near ''z'' = 1. This power series can be turned into a Germ

g

This germ has a radius of convergence of 1, and so there is a Sheaf ''S'' corresponding to it. 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 Function s''. See Sheaf for the general concept.

MONODROMY THEOREM

See Also: Monodromy theorem

The monodromy theorem gives a sufficient condition for the existence of a ''direct analytic continuation'' (i.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 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

See Also: 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	o\infty} rac{k(i+1)}{k(i)} > 1 + \delta \,

for some fixed δ > 0, the circle centre ''z''₀ and with radius the radius of convergence is a natural boundary. Such a power series defines a Lacunary Function .

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 ''z''₀ as a natural boundary.

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

SEE ALSO

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Analytic Continuation