Activity 1.
Let
 be
an analytic function in a domain D. Let such
that
 .
Prove that
 on
D or
 N
and analytic function
in D such that
 with
 .
(Hint: Use the analytic continuation principle.)
Activity 2.
Prove L’Hopital’s rule for complex variables: Let
and
be analytic functions in a domain D and not
identically 0. If
and
 at
some point
,
then
  .
(Hint:
Use Activity 1.)
Activity
3.
Let
be
an analytic function in
and let
with
.
If
 for
show that
.
(Hint:
Use
Activity 1 and apply the maximum principle to
and
 .) |
