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Analytic continuation

Analytic continuation. Let f_{1}(z) be a
function that is analytic in a region R_{1} of the
complex plane and f_{2}(z) a function that is
analytic in a region R_{2} partly overlapping R_{1}.
If f_{1}(z) = f_{2}(z) in the overlapping part, then
f_{2}(z) is called the analytic continuation of
f_{1}(z). This means that there is a function f(z)
that is analytic in the combined regions R_{1}
and R_{2 }such that f(z) = f_{1}(z) in R_{1} and f(z) =
f_{2}(z) in R_{2}. It is sufficient for R_{1} and R_{2} to
have only a small arc in common such as the
arc ABC shown in Fig. 2.

By analytic continuation to regions R_{3}, R_{4, }etc. we can extend the original region of definition to
other parts of the complex plane. The
functions f_{1}(z), f_{2}(z), f_{3}(z), ... defined in R_{1},
R_{2}, R_{3,} ... respectively, are called function
elements or briefly elements. It is
sometimes impossible to extend a function
analytically beyond the boundary of a
region. In such a case the boundary is
called a natural boundary.

Uniqueness theorem for analytic
continuation. Let a function f_{1}(z)
defined in R_{1} be continued analytically to
region R_{n} along two different paths. See
Fig. 3. Then the two analytic continuations
will be identical providing there is no
singularity between the paths.

If we do get different results when using two different paths we can show that there is a singularity (specifically a branch point) between the paths.

One can illustrate analytic continuation
with Taylor series expansions. Suppose
we do not know the exact form of an
analytic function f(z) but only know that
inside some circle of convergence C_{1}
with center at a f(z) is represented by a
Taylor series

1) a_{0} + a_{1}(z - a) + a_{2}(z - a)^{2} + ...

If we then choose a point b inside C_{1} we can
find the value of f(z) and its derivatives at b
from 1) and thus arrive at a new series

2) b_{0} + b_{1}(z - b) + b_{2}(z - b)^{2} + ...

having a circle of convergence C_{2}. See Fig. 4.
We can then choose a point c inside C_{2} and
repeat the process. The process can be
repeated indefinitely.

The collection of all such power series representations, i.e. all possible analytic continuations, is defined as the analytic function f(z).

References

Spiegel. Complex Variables. (Schaum)

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