and
be
analytic functions whose domains are the regions
and
respectively.andare
direct analytic continuations of each other if there is a region
such
that
for
University Maths Notes: Complex Analysis – Analytic Continuation
We start with a
Definition Letandbe
analytic functions whose domains are the regionsand
respectively.andare
direct analytic continuations of each other if there is a regionsuch
thatfor
Example:
is
only defined for
but
we may write
on
the region defined by
but
is
defined on the wider region
is
an analytic continuation of
Example:
is
normally defined on the region
so
that
We in fact only need Log to be one to one on the complex plane so
that the inverse is defined. We can instead define Log_1 z on
so that
for
and
we have extended
to
be defined on
