Exercises

Sessions

 

Exercise 1.  Let be an open convex set in the plane and let be a function holomorphic in   Show that, for each  pair of points we can find two points and on the segment joining and such that

(Hint: Consider the function of a real variable defined by

= and apply the Mean Value Theorem to the real and imaginary parts of

 

Exercise 2.   Prove the following :

Let a be a limit point of a subset D of C.  Suppose that converges uniformly to f on D

 and that   Then converges and ; that is,

Exercise 3.   Prove the following :

Suppose that is a sequence of integrable functions on [a,b] which converges

uniformly to f on [a,b].  Then f is integrable and

Exercise 4.  Test for uniform convergence in the given region:

a) 

b) 

c) 

Exercise 5.   Show that