Exercises

Sessions

 

Exercise 1.  Prove Greenís 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: