Exercise 1.
Prove Green’s Theorem:
Suppose that D is a domain with boundary C both
contained in an open set S on which f has
continuous partial derivatives. Then
Exercise 2.
Let f and g be two holomorphic functions on
a domain
of the plane, not identically zero on
; if there exists a sequence {} of points in D such that
as
with
and
for all
, show that there exists a constant
such that
on
Exercise 3.
Let
be a continuous function on the
oriented boundary
of a compact set
Let
be the open complement of
in
C.
For
put
i) Put
for
Show that, for
and
with
can be expanded as a normally
convergent power series in terms of
deduce that
is analytic in a neighborhood of each
ii) Show that
, for all integers
Exercise 4.
Let
be holomorphic in
show that if
then
as
with
uniformly on
(Hint: Using
exercise 3, write
where
for example deduce that if
then
Exercise 5.
Let
be holomorphic in
Evaluate, in two different methods,
the integral
(2())
with the unit circle oriented in the direct sense and
deduce the following equalities:
