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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:
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