          Exercises Sessions  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:   