In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent.
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 singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology.
Suppose f is an analytic function defined on a non-empty open subset U of the complex plane If V is a larger open subset of containing U, and F is an analytic function defined on V such that
F(z)=f(z) \forallz\inU,
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 the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U
F1(z)=F2(z)=f(z),
then
F1=F2
on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions.
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. Examples are the Riemann zeta function and the gamma function.
The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces.
Analytic continuation is used in Riemannian manifolds, solutions of Einstein's equations. For example, the analytic continuation of Schwarzschild coordinates into Kruskal–Szekeres coordinates.[1]
Begin with a particular analytic function
f
z=1
By the Cauchy–Hadamard theorem, its radius of convergence is 1. That is,
f
U=\{|z-1|<1\}
\partialU=\{|z-1|=1\}
z=0\in\partialU
Pretend we don't know that
f(z)=1/z
a\inU
We'll calculate the
ak
V
U
f
U\cupV
U
The distance from
a
\partialU
\rho=1-|a-1|>0
0<r<\rho
D
r
a
\partialD
D\cup\partialD\subsetU
The last summation results from the th derivation of the geometric series, which gives the formula
Then,
which has radius of convergence
|a|
0
a\inU
|a|>1
V
U
U
a=\tfrac{1}{2}(3+i).
We can continue the process: select
b\inU\cupV
b
U\cupV
f
f
\Complex\setminus\{0\}.
In this particular case the obtained values of
f(-1)
The power series defined below is generalized by the idea of a germ. The general theory of analytic continuation and its generalizations is known as sheaf theory. Let
infty | |
f(z)=\sum | |
k=0 |
\alphak
k | |
(z-z | |
0) |
be a power series converging in the disk Dr(z0), r > 0, defined by
Dr(z0)=\{z\in\Complex:|z-z0|<r\}
Note that without loss of generality, here and below, we will always assume that a maximal such r was chosen, even if that r 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,\alpha0,\alpha1,\alpha2,\ldots)
is a germ of f. 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.
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
lG
Let g and h be germs. If
|h0-g0|<r
\cong
We can define a topology on
lG
Ur(g)=\{h\inlG:g\geh,|g0-h0|<r\}.
The sets Ur(g), for all r > 0 and
g\inlG
lG
A connected component of
lG
\phig(h)=h0:Ur(g)\to\Complex,
lG
lG
lG
L(z)=
infin | |
\sum | |
k=1 |
(-1)k+1 | |
k |
(z-1)k
is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ
g=\left(1,0,1,- | 1 |
2, |
1 | |
3 |
,-
1 | |
4 |
,
1 | |
5 |
,-
1 | |
6 |
,\ldots\right)
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 functions. See sheaf for the general concept.
Suppose that a power series has radius of convergence r and defines an analytic function f inside that disc. Consider points on the circle of convergence. A point for which there is a neighbourhood on which f has an analytic extension is regular, otherwise singular. The circle is a natural boundary if all its points are singular.
More generally, we may apply the definition to any open connected domain on which f is analytic, and classify the points of the boundary of the domain as regular or singular: the domain boundary is then a natural boundary if all points are singular, in which case the domain is a domain of holomorphy.
For
\Re(s)>1
P(s)
P(s):=\sump primep-s.
This function is analogous to the summatory form of the Riemann zeta function when
\Re(s)>1
\zeta(s)
0<\Re(s)<1
P(s)
P(s)=\sumn\mu(n)
log\zeta(ns) | |
n |
.
Since
\zeta(s)
s:=1
P(s)
s:=\tfrac{1}{k},\forallk\in\Z+
\operatorname{Sing}P:=\left\{k-1:k\in\Z+\right\}=\left\{1,
1 | |
2 |
,
1 | |
3 |
,
1 | |
4 |
,\ldots\right\}
has accumulation point 0 (the limit of the sequence as
k\mapstoinfty
P(s)
P(s)
P(s)
0\geq\Re(s)
IF\subseteq\Complex suchthat \Re(s)\in(-C,C),\foralls\inIF
C>0
P(s)
For integers
c\geq2
l{L}c(z):=\sumn
cn | |
z |
,|z|<1.
Clearly, since
cn+1=c ⋅ cn
l{L}c(z)
|z|<1
l{L}c(z)=zc+
c) | |
l{L} | |
c(z |
m\geq1
l{L}c(z)
l{L}c(z)=
m-1 | |
\sum | |
i=0 |
ci | |
z |
+
cm | |
l{L} | |
c(z |
),\forall|z|<1.
For any positive natural numbers c, the lacunary series function diverges at
z=1
l{L}c(z)
|z|>1.
n\geq1
l{L}c(z)
cn
l{L}c(z)
The proof of this fact is generalized from a standard argument for the case where
c:=2.
n\geq1
l{R}c,n:=\left\{z\inD\cup\partial{D
where
D
|l{R}c,n|=cn
cn
cn | |
z |
=1
l{L}c(z)
|z|<1
\forallz\inl{R}c,n, l{L}c(z)=
cn-1 | |
\sum | |
i=0 |
ci | |
z |
+
cn | |
l{L} | |
c(z |
)=
cn-1 | |
\sum | |
i=0 |
ci | |
z |
+l{L}c(1)=+infty.
Thus for any arc on the boundary of the unit circle, there are an infinite number of points z within this arc such that
l{L}c(z)=infty
C1:=\{z:|z|=1\}
l{L}c(z)
c\in\Z c>1.
See main article: 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\subset\Complex
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.
See main article: Ostrowski–Hadamard gap theorem.
For a power series
infty | |
f(z)=\sum | |
k=0 |
ak
nk | |
z |
with
\liminfk\toinfty
nk+1 | |
nk |
>1
the circle of convergence is a natural boundary. Such a power series is called lacunary.This theorem has been substantially generalized by Eugen Fabry (see Fabry's gap theorem) and George Pólya.
Let
infty | |
f(z)=\sum | |
k=0 |
\alphak
k | |
(z-z | |
0) |
be a power series, then there exist εk ∈ such that
infty | |
f(z)=\sum | |
k=0 |
\varepsilonk\alphak
k | |
(z-z | |
0) |
has the convergence disc of f around z0 as a natural boundary.
The proof of this theorem makes use of Hadamard's gap theorem.