Exercise 1.  Let  be a complex parameter. 

i) Show that the Laurent series expansion of the function exp() at the origin  has the form

exp()= for  with   for  

Similarly, show that the Laurent series expansion of the function exp() at the origin  has the form

exp()= for  with   for  

(Notice that if =- then exp()=exp() for  

ii) Let  and  be two nonnegative integers.  Show that


 if with  being a nonnegative integer and  otherwise, and deduce the expansions in power series of and  as functions of the parameter  ( , as a function of  carries the name of a Bessel function of the first kind).

 

Exercise 2.    Let  be a meromorphic function in a neighborhood of the origin  and having a simple pole at the origin.  Let  be an arbitrary complex number.  Show that the Laurent expansion of the function  has the form

- where  is a polynomial in  of degree  (One can make an identification by using the Taylor expansion of the function  

 

Exercise 3.    Let  be a holomorphic function in the upper half plane  defined by   suppose  for    Show that there exists a holomorphic function  on the punctured disk  such that  for    Deduce that  has the expansion

 with  for some .

Show that this series  converges uniformly on each compact subset of   Show that if there exists a constant  and an integer  such that 

    for all sufficiently large, and uniformly in then  

Exercise 4.    i)  Show that the function  is meromorphic on the whole complex plane and has the points    as simple poles.  Find the Laurent expansion at the point   If  ()  denote the coefficients of the expansion for   show that   for an integer  and if we put

  =( ()  ! for  show that we have the recurrence formula:

 -  +     for   (by identification of the coefficients on the two sides of the relation       ).

ii)  For  put   Let   be the perimeter of a square having vertices

 Show that if  is on  , then  

Integrating  along the contour  in the direct sense and making  deduce that

 

(N.B. The numbers  are called the Bernoulli numbers)

 

 Exercise 5.   Find the Laurent series of  in

a) 

b) 

 

Exercise 6.    Find the Laurent series of  in

a) 

b) 

c) 

d)