Exercise 1.   Solve the equation

a)       

b)      .

c)      Is the function f(z)=onto? How about f(z)=?

d)      Is the function f(z)=onto?

Exercise 2.   Let  and  be real numbers with   integer  Let  be a non-negative integer.  Show that

(Use  

Exercise 3.    Show that  for all  

Exercise 4.  Show that  for all integers  and for all C and deduce that  for all  C.

Exercise 5.  Show that the function  respectively  is the analytic continuation, in  , of the function respectively  

Show that

where  and  are in C.

Exercise 6.   Show that  for  

Exercise 7.   Let  

i)  Show that

ii) Determine the zeros of the functions  and , where

 is a non-zero real number.

iii)  If  and  is a positive integer, show that

 for  

and

 for  

(Hint: Using definitions, ).

Exercise 8.    Let  be an interval in ,  If  is a real-valued function with values in C which is analytic in I, then we can extend it to an analytic function in an open and connected subset of  C containing  

Exercise 9.    i)  Let  and  be sequences s.t.

a) There exists  s.t.  for each  

b)  and  

Show that , for each  

(Introduce  and write

ii) Let  be a  power series with complex coefficients s.t.  and s.t.  is convergent.  Use (i) to show that  converges uniformly on  and conclude that with   .

iii)  Now let  and  be the intersection of the disk  and the disk   Show that there exists a constant  s.t.

 for  where  denotes the principal determination of the complex logarithmic function defined on the half-plane  which contains  

(Note that if  then  with  and

 for  

Use (ii) to show that

and