Analytic continuation is the extension of an analytic function beyond its initial domain by agreeing analytic functions on overlapping domains. It can be carried out by a chain of power series centered at successive points of a path. An initial power series
 |
(1)
|
with radius of convergence 
can be reexpanded about 
when 
. The new coefficients
are
 |
(2)
|
where 
is a nonnegative integer and 
is a binomial coefficient.
The new expansion is 
at least for 
. The new radius
of convergence can exceed this bound, so the new power
series may reach points outside the original disk. Iterating
this transformation along a path is the power
series method of analytic continuation.
For example, 
has Taylor series 
about the origin with radius
of convergence 1. Reexpansion about 
gives
 |
(3)
|
which converges for 
and therefore includes points outside 
.
If 
and 
are analytic functions on domains 
and 
and agree on their nonempty intersection,
then 
is an analytic continuation of 
, and vice versa (Flanigan 1983, p. 234). The principle
of permanence makes such a direct
analytic continuation to a specified connected domain unique when it exists. Along a fixed path,
continuation from a specified initial function element
is likewise unique. Different paths to the same point can
give different branches, as described by monodromy.
The monodromy theorem gives conditions under
which the result is independent of the path.
Continuation need not be possible along every path. A singularity can prevent it, and a natural boundary can prevent extension across every point of a boundary. Thus, local values determine any continuation uniquely along a specified path, but do not guarantee a single-valued function on the entire complex plane.
Analytic continuation can lead to some interesting phenomena such as multivalued functions. For example, consider analytic continuation of the square
root function 
. Although this function is not globally well-defined
(since every nonzero complex number has two square roots), 
has a well-defined Taylor series
around 
,
 |  |  |
(4)
|
 |  |  |
(5)
|
which has radius of convergence 1 about 
and converges in 
.
The animation above shows the analytic continuation of 
along the path 
. Note that when the function goes all the way around,
is the negative of the original function, so going around twice returns the function
to its original value.
In the animation, the domain space (colored pink; left figures) is mapped to the image space (colored blue; right figures) by the square root function, and the light blue region indicates the negative square root. However, by continuing the function around the circle, the square root function takes values in what used to be the light blue region, so the roles of the blue and light blue region are reversed.
This can be interpreted as going from one branch of the multivalued square root function to the other. This illustrates that analytic continuation extends a function using the nearby values that provide the information on the power series.
It is possible for the function to never return to the same value. For example, 
increases by 
every time it is continued counterclockwise around zero.
This gives a logarithmic branch point.
The natural domain is described by a maximal chain
of domains on which the continued function is a single-valued
function. Its Riemann surface is the universal
cover of the punctured plane for 
and the connected double cover
for 
.
If there is a boundary across which the function cannot be extended, it is called
a natural boundary. For instance, there exists
a meromorphic function 
in the unit disk where every
point on the unit circle is a limit
point of the set of poles. Then the circle
is a natural boundary for 
.
